Before switching to generic theory of relational trees.
section {* This file contains lemmas used to guide the recalculation of current precedence after every system call (or system operation)*}theory CpsGimports PrioG Max RTreebeginlocale pip = fixes s assumes vt: "vt s"context pipbegininterpretation rtree_RAG: rtree "RAG s"proof show "single_valued (RAG s)" apply (intro_locales) by (unfold single_valued_def, auto intro: unique_RAG[OF vt]) show "acyclic (RAG s)" by (rule acyclic_RAG[OF vt])qedenddefinition "the_preced s th = preced th s"lemma cp_alt_def: "cp s th = Max ((the_preced s) ` {th'. Th th' \<in> (subtree (RAG s) (Th th))})"proof - have "Max (the_preced s ` ({th} \<union> dependants (wq s) th)) = Max (the_preced s ` {th'. Th th' \<in> subtree (RAG s) (Th th)})" (is "Max (_ ` ?L) = Max (_ ` ?R)") proof - have "?L = ?R" by (auto dest:rtranclD simp:cs_dependants_def cs_RAG_def s_RAG_def subtree_def) thus ?thesis by simp qed thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp)qedlemma eq_dependants: "dependants (wq s) = dependants s" by (simp add: s_dependants_abv wq_def)(* obvious lemma *)lemma not_thread_holdents: fixes th s assumes vt: "vt s" and not_in: "th \<notin> threads s" shows "holdents s th = {}"proof - from vt not_in show ?thesis proof(induct arbitrary:th) case (vt_cons s e th) assume vt: "vt s" and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> holdents s th = {}" and stp: "step s e" and not_in: "th \<notin> threads (e # s)" from stp show ?case proof(cases) case (thread_create thread prio) assume eq_e: "e = Create thread prio" and not_in': "thread \<notin> threads s" have "holdents (e # s) th = holdents s th" apply (unfold eq_e holdents_test) by (simp add:RAG_create_unchanged) moreover have "th \<notin> threads s" proof - from not_in eq_e show ?thesis by simp qed moreover note ih ultimately show ?thesis by auto next case (thread_exit thread) assume eq_e: "e = Exit thread" and nh: "holdents s thread = {}" show ?thesis proof(cases "th = thread") case True with nh eq_e show ?thesis by (auto simp:holdents_test RAG_exit_unchanged) next case False with not_in and eq_e have "th \<notin> threads s" by simp from ih[OF this] False eq_e show ?thesis by (auto simp:holdents_test RAG_exit_unchanged) qed next case (thread_P thread cs) assume eq_e: "e = P thread cs" and is_runing: "thread \<in> runing s" from assms thread_exit ih stp not_in vt eq_e have vtp: "vt (P thread cs#s)" by auto have neq_th: "th \<noteq> thread" proof - from not_in eq_e have "th \<notin> threads s" by simp moreover from is_runing have "thread \<in> threads s" by (simp add:runing_def readys_def) ultimately show ?thesis by auto qed hence "holdents (e # s) th = holdents s th " apply (unfold cntCS_def holdents_test eq_e) by (unfold step_RAG_p[OF vtp], auto) moreover have "holdents s th = {}" proof(rule ih) from not_in eq_e show "th \<notin> threads s" by simp qed ultimately show ?thesis by simp next case (thread_V thread cs) assume eq_e: "e = V thread cs" and is_runing: "thread \<in> runing s" and hold: "holding s thread cs" have neq_th: "th \<noteq> thread" proof - from not_in eq_e have "th \<notin> threads s" by simp moreover from is_runing have "thread \<in> threads s" by (simp add:runing_def readys_def) ultimately show ?thesis by auto qed from assms thread_V eq_e ih stp not_in vt have vtv: "vt (V thread cs#s)" by auto from hold obtain rest where eq_wq: "wq s cs = thread # rest" by (case_tac "wq s cs", auto simp: wq_def s_holding_def) from not_in eq_e eq_wq have "\<not> next_th s thread cs th" apply (auto simp:next_th_def) proof - assume ne: "rest \<noteq> []" and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> threads s" (is "?t \<notin> threads s") have "?t \<in> set rest" proof(rule someI2) from wq_distinct[OF step_back_vt[OF vtv], of cs] and eq_wq show "distinct rest \<and> set rest = set rest" by auto next fix x assume "distinct x \<and> set x = set rest" with ne show "hd x \<in> set rest" by (cases x, auto) qed with eq_wq have "?t \<in> set (wq s cs)" by simp from wq_threads[OF step_back_vt[OF vtv], OF this] and ni show False by auto qed moreover note neq_th eq_wq ultimately have "holdents (e # s) th = holdents s th" by (unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto) moreover have "holdents s th = {}" proof(rule ih) from not_in eq_e show "th \<notin> threads s" by simp qed ultimately show ?thesis by simp next case (thread_set thread prio) print_facts assume eq_e: "e = Set thread prio" and is_runing: "thread \<in> runing s" from not_in and eq_e have "th \<notin> threads s" by auto from ih [OF this] and eq_e show ?thesis apply (unfold eq_e cntCS_def holdents_test) by (simp add:RAG_set_unchanged) qed next case vt_nil show ?case by (auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def) qedqed(* obvious lemma *)lemma next_th_neq: assumes vt: "vt s" and nt: "next_th s th cs th'" shows "th' \<noteq> th"proof - from nt show ?thesis apply (auto simp:next_th_def) proof - fix rest assume eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest" and ne: "rest \<noteq> []" have "hd (SOME q. distinct q \<and> set q = set rest) \<in> set rest" proof(rule someI2) from wq_distinct[OF vt, of cs] eq_wq show "distinct rest \<and> set rest = set rest" by auto next fix x assume "distinct x \<and> set x = set rest" hence eq_set: "set x = set rest" by auto with ne have "x \<noteq> []" by auto hence "hd x \<in> set x" by auto with eq_set show "hd x \<in> set rest" by auto qed with wq_distinct[OF vt, of cs] eq_wq show False by auto qedqed(* obvious lemma *)lemma next_th_unique: assumes nt1: "next_th s th cs th1" and nt2: "next_th s th cs th2" shows "th1 = th2"using assms by (unfold next_th_def, auto)lemma wf_RAG: assumes vt: "vt s" shows "wf (RAG s)"proof(rule finite_acyclic_wf) from finite_RAG[OF vt] show "finite (RAG s)" .next from acyclic_RAG[OF vt] show "acyclic (RAG s)" .qeddefinition child :: "state \<Rightarrow> (node \<times> node) set" where "child s \<equiv> {(Th th', Th th) | th th'. \<exists>cs. (Th th', Cs cs) \<in> RAG s \<and> (Cs cs, Th th) \<in> RAG s}"definition children :: "state \<Rightarrow> thread \<Rightarrow> thread set" where "children s th \<equiv> {th'. (Th th', Th th) \<in> child s}"lemma children_def2: "children s th \<equiv> {th'. \<exists> cs. (Th th', Cs cs) \<in> RAG s \<and> (Cs cs, Th th) \<in> RAG s}"unfolding child_def children_def by simplemma children_dependants: "children s th \<subseteq> dependants (wq s) th" unfolding children_def2 unfolding cs_dependants_def by (auto simp add: eq_RAG)lemma child_unique: assumes vt: "vt s" and ch1: "(Th th, Th th1) \<in> child s" and ch2: "(Th th, Th th2) \<in> child s" shows "th1 = th2"using ch1 ch2 proof(unfold child_def, clarsimp) fix cs csa assume h1: "(Th th, Cs cs) \<in> RAG s" and h2: "(Cs cs, Th th1) \<in> RAG s" and h3: "(Th th, Cs csa) \<in> RAG s" and h4: "(Cs csa, Th th2) \<in> RAG s" from unique_RAG[OF vt h1 h3] have "cs = csa" by simp with h4 have "(Cs cs, Th th2) \<in> RAG s" by simp from unique_RAG[OF vt h2 this] show "th1 = th2" by simpqed lemma RAG_children: assumes h: "(Th th1, Th th2) \<in> (RAG s)^+" shows "th1 \<in> children s th2 \<or> (\<exists> th3. th3 \<in> children s th2 \<and> (Th th1, Th th3) \<in> (RAG s)^+)"proof - from h show ?thesis proof(induct rule: tranclE) fix c th2 assume h1: "(Th th1, c) \<in> (RAG s)\<^sup>+" and h2: "(c, Th th2) \<in> RAG s" from h2 obtain cs where eq_c: "c = Cs cs" by (case_tac c, auto simp:s_RAG_def) show "th1 \<in> children s th2 \<or> (\<exists>th3. th3 \<in> children s th2 \<and> (Th th1, Th th3) \<in> (RAG s)\<^sup>+)" proof(rule tranclE[OF h1]) fix ca assume h3: "(Th th1, ca) \<in> (RAG s)\<^sup>+" and h4: "(ca, c) \<in> RAG s" show "th1 \<in> children s th2 \<or> (\<exists>th3. th3 \<in> children s th2 \<and> (Th th1, Th th3) \<in> (RAG s)\<^sup>+)" proof - from eq_c and h4 obtain th3 where eq_ca: "ca = Th th3" by (case_tac ca, auto simp:s_RAG_def) from eq_ca h4 h2 eq_c have "th3 \<in> children s th2" by (auto simp:children_def child_def) moreover from h3 eq_ca have "(Th th1, Th th3) \<in> (RAG s)\<^sup>+" by simp ultimately show ?thesis by auto qed next assume "(Th th1, c) \<in> RAG s" with h2 eq_c have "th1 \<in> children s th2" by (auto simp:children_def child_def) thus ?thesis by auto qed next assume "(Th th1, Th th2) \<in> RAG s" thus ?thesis by (auto simp:s_RAG_def) qedqedlemma sub_child: "child s \<subseteq> (RAG s)^+" by (unfold child_def, auto)lemma wf_child: assumes vt: "vt s" shows "wf (child s)"apply(rule wf_subset)apply(rule wf_trancl[OF wf_RAG[OF vt]])apply(rule sub_child)donelemma RAG_child_pre: assumes vt: "vt s" shows "(Th th, n) \<in> (RAG s)^+ \<longrightarrow> (\<forall> th'. n = (Th th') \<longrightarrow> (Th th, Th th') \<in> (child s)^+)" (is "?P n")proof - from wf_trancl[OF wf_RAG[OF vt]] have wf: "wf ((RAG s)^+)" . show ?thesis proof(rule wf_induct[OF wf, of ?P], clarsimp) fix th' assume ih[rule_format]: "\<forall>y. (y, Th th') \<in> (RAG s)\<^sup>+ \<longrightarrow> (Th th, y) \<in> (RAG s)\<^sup>+ \<longrightarrow> (\<forall>th'. y = Th th' \<longrightarrow> (Th th, Th th') \<in> (child s)\<^sup>+)" and h: "(Th th, Th th') \<in> (RAG s)\<^sup>+" show "(Th th, Th th') \<in> (child s)\<^sup>+" proof - from RAG_children[OF h] have "th \<in> children s th' \<or> (\<exists>th3. th3 \<in> children s th' \<and> (Th th, Th th3) \<in> (RAG s)\<^sup>+)" . thus ?thesis proof assume "th \<in> children s th'" thus "(Th th, Th th') \<in> (child s)\<^sup>+" by (auto simp:children_def) next assume "\<exists>th3. th3 \<in> children s th' \<and> (Th th, Th th3) \<in> (RAG s)\<^sup>+" then obtain th3 where th3_in: "th3 \<in> children s th'" and th_dp: "(Th th, Th th3) \<in> (RAG s)\<^sup>+" by auto from th3_in have "(Th th3, Th th') \<in> (RAG s)^+" by (auto simp:children_def child_def) from ih[OF this th_dp, of th3] have "(Th th, Th th3) \<in> (child s)\<^sup>+" by simp with th3_in show "(Th th, Th th') \<in> (child s)\<^sup>+" by (auto simp:children_def) qed qed qedqedlemma RAG_child: "\<lbrakk>vt s; (Th th, Th th') \<in> (RAG s)^+\<rbrakk> \<Longrightarrow> (Th th, Th th') \<in> (child s)^+" by (insert RAG_child_pre, auto)lemma child_RAG_p: assumes "(n1, n2) \<in> (child s)^+" shows "(n1, n2) \<in> (RAG s)^+"proof - from assms show ?thesis proof(induct) case (base y) with sub_child show ?case by auto next case (step y z) assume "(y, z) \<in> child s" with sub_child have "(y, z) \<in> (RAG s)^+" by auto moreover have "(n1, y) \<in> (RAG s)^+" by fact ultimately show ?case by auto qedqedtext {* (* ddd *)*}lemma child_RAG_eq: assumes vt: "vt s" shows "(Th th1, Th th2) \<in> (child s)^+ \<longleftrightarrow> (Th th1, Th th2) \<in> (RAG s)^+" by (auto intro: RAG_child[OF vt] child_RAG_p)text {* (* ddd *)*}lemma children_no_dep: fixes s th th1 th2 th3 assumes vt: "vt s" and ch1: "(Th th1, Th th) \<in> child s" and ch2: "(Th th2, Th th) \<in> child s" and ch3: "(Th th1, Th th2) \<in> (RAG s)^+" shows "False"proof - from RAG_child[OF vt ch3] have "(Th th1, Th th2) \<in> (child s)\<^sup>+" . thus ?thesis proof(rule converse_tranclE) assume "(Th th1, Th th2) \<in> child s" from child_unique[OF vt ch1 this] have "th = th2" by simp with ch2 have "(Th th2, Th th2) \<in> child s" by simp with wf_child[OF vt] show ?thesis by auto next fix c assume h1: "(Th th1, c) \<in> child s" and h2: "(c, Th th2) \<in> (child s)\<^sup>+" from h1 obtain th3 where eq_c: "c = Th th3" by (unfold child_def, auto) with h1 have "(Th th1, Th th3) \<in> child s" by simp from child_unique[OF vt ch1 this] have eq_th3: "th3 = th" by simp with eq_c and h2 have "(Th th, Th th2) \<in> (child s)\<^sup>+" by simp with ch2 have "(Th th, Th th) \<in> (child s)\<^sup>+" by auto moreover have "wf ((child s)\<^sup>+)" proof(rule wf_trancl) from wf_child[OF vt] show "wf (child s)" . qed ultimately show False by auto qedqedtext {* (* ddd *)*}lemma unique_RAG_p: assumes vt: "vt s" and dp1: "(n, n1) \<in> (RAG s)^+" and dp2: "(n, n2) \<in> (RAG s)^+" and neq: "n1 \<noteq> n2" shows "(n1, n2) \<in> (RAG s)^+ \<or> (n2, n1) \<in> (RAG s)^+"proof(rule unique_chain [OF _ dp1 dp2 neq]) from unique_RAG[OF vt] show "\<And>a b c. \<lbrakk>(a, b) \<in> RAG s; (a, c) \<in> RAG s\<rbrakk> \<Longrightarrow> b = c" by autoqedtext {* (* ddd *)*}lemma dependants_child_unique: fixes s th th1 th2 th3 assumes vt: "vt s" and ch1: "(Th th1, Th th) \<in> child s" and ch2: "(Th th2, Th th) \<in> child s" and dp1: "th3 \<in> dependants s th1" and dp2: "th3 \<in> dependants s th2"shows "th1 = th2"proof - { assume neq: "th1 \<noteq> th2" from dp1 have dp1: "(Th th3, Th th1) \<in> (RAG s)^+" by (simp add:s_dependants_def eq_RAG) from dp2 have dp2: "(Th th3, Th th2) \<in> (RAG s)^+" by (simp add:s_dependants_def eq_RAG) from unique_RAG_p[OF vt dp1 dp2] and neq have "(Th th1, Th th2) \<in> (RAG s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (RAG s)\<^sup>+" by auto hence False proof assume "(Th th1, Th th2) \<in> (RAG s)\<^sup>+ " from children_no_dep[OF vt ch1 ch2 this] show ?thesis . next assume " (Th th2, Th th1) \<in> (RAG s)\<^sup>+" from children_no_dep[OF vt ch2 ch1 this] show ?thesis . qed } thus ?thesis by autoqedlemma RAG_plus_elim: assumes "vt s" fixes x assumes "(Th x, Th th) \<in> (RAG (wq s))\<^sup>+" shows "\<exists>th'\<in>children s th. x = th' \<or> (Th x, Th th') \<in> (RAG (wq s))\<^sup>+" using assms(2)[unfolded eq_RAG, folded child_RAG_eq[OF `vt s`]] apply (unfold children_def) by (metis assms(2) children_def RAG_children eq_RAG)text {* (* ddd *)*}lemma dependants_expand: assumes "vt s" shows "dependants (wq s) th = (children s th) \<union> (\<Union>((dependants (wq s)) ` children s th))"apply(simp add: image_def)unfolding cs_dependants_defapply(auto)apply (metis assms RAG_plus_elim mem_Collect_eq)apply (metis child_RAG_p children_def eq_RAG mem_Collect_eq r_into_trancl')by (metis assms child_RAG_eq children_def eq_RAG mem_Collect_eq trancl.trancl_into_trancl)lemma finite_children: assumes "vt s" shows "finite (children s th)" using children_dependants dependants_threads[OF assms] finite_threads[OF assms] by (metis rev_finite_subset)lemma finite_dependants: assumes "vt s" shows "finite (dependants (wq s) th')" using dependants_threads[OF assms] finite_threads[OF assms] by (metis rev_finite_subset)abbreviation "preceds s ths \<equiv> {preced th s| th. th \<in> ths}"abbreviation "cpreceds s ths \<equiv> (cp s) ` ths"lemma Un_compr: "{f th | th. R th \<or> Q th} = ({f th | th. R th} \<union> {f th' | th'. Q th'})"by autolemma in_disj: shows "x \<in> A \<or> (\<exists>y \<in> A. x \<in> Q y) \<longleftrightarrow> (\<exists>y \<in> A. x = y \<or> x \<in> Q y)"by metislemma UN_exists: shows "(\<Union>x \<in> A. {f y | y. Q y x}) = ({f y | y. (\<exists>x \<in> A. Q y x)})"by autotext {* (* ddd *) This is the recursive equation used to compute the current precedence of a thread (the @{text "th"}) here. *}lemma cp_rec: fixes s th assumes vt: "vt s" shows "cp s th = Max ({preced th s} \<union> (cp s ` children s th))"proof(cases "children s th = {}") case True show ?thesis unfolding cp_eq_cpreced cpreced_def by (subst dependants_expand[OF `vt s`]) (simp add: True)next case False show ?thesis (is "?LHS = ?RHS") proof - have eq_cp: "cp s = (\<lambda>th. Max (preceds s ({th} \<union> dependants (wq s) th)))" by (simp add: fun_eq_iff cp_eq_cpreced cpreced_def Un_compr image_Collect[symmetric]) have not_emptyness_facts[simp]: "dependants (wq s) th \<noteq> {}" "children s th \<noteq> {}" using False dependants_expand[OF assms] by(auto simp only: Un_empty) have finiteness_facts[simp]: "\<And>th. finite (dependants (wq s) th)" "\<And>th. finite (children s th)" by (simp_all add: finite_dependants[OF `vt s`] finite_children[OF `vt s`]) (* expanding definition *) have "?LHS = Max ({preced th s} \<union> preceds s (dependants (wq s) th))" unfolding eq_cp by (simp add: Un_compr) (* moving Max in *) also have "\<dots> = max (Max {preced th s}) (Max (preceds s (dependants (wq s) th)))" by (simp add: Max_Un) (* expanding dependants *) also have "\<dots> = max (Max {preced th s}) (Max (preceds s (children s th \<union> \<Union>(dependants (wq s) ` children s th))))" by (subst dependants_expand[OF `vt s`]) (simp) (* moving out big Union *) also have "\<dots> = max (Max {preced th s}) (Max (preceds s (\<Union> ({children s th} \<union> (dependants (wq s) ` children s th)))))" by simp (* moving in small union *) also have "\<dots> = max (Max {preced th s}) (Max (preceds s (\<Union> ((\<lambda>th. {th} \<union> (dependants (wq s) th)) ` children s th))))" by (simp add: in_disj) (* moving in preceds *) also have "\<dots> = max (Max {preced th s}) (Max (\<Union> ((\<lambda>th. preceds s ({th} \<union> (dependants (wq s) th))) ` children s th)))" by (simp add: UN_exists) (* moving in Max *) also have "\<dots> = max (Max {preced th s}) (Max ((\<lambda>th. Max (preceds s ({th} \<union> (dependants (wq s) th)))) ` children s th))" by (subst Max_Union) (auto simp add: image_image) (* folding cp + moving out Max *) also have "\<dots> = ?RHS" unfolding eq_cp by (simp add: Max_insert) finally show "?LHS = ?RHS" . qedqedlemma next_waiting: assumes vt: "vt s" and nxt: "next_th s th cs th'" shows "waiting s th' cs"proof - from assms show ?thesis apply (auto simp:next_th_def s_waiting_def[folded wq_def]) proof - fix rest assume ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set rest" and eq_wq: "wq s cs = th # rest" and ne: "rest \<noteq> []" have "set (SOME q. distinct q \<and> set q = set rest) = set rest" proof(rule someI2) from wq_distinct[OF vt, of cs] eq_wq show "distinct rest \<and> set rest = set rest" by auto next show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto qed with ni have "hd (SOME q. distinct q \<and> set q = set rest) \<notin> set (SOME q. distinct q \<and> set q = set rest)" by simp moreover have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []" proof(rule someI2) from wq_distinct[OF vt, of cs] eq_wq show "distinct rest \<and> set rest = set rest" by auto next from ne show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> x \<noteq> []" by auto qed ultimately show "hd (SOME q. distinct q \<and> set q = set rest) = th" by auto next fix rest assume eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest" and ne: "rest \<noteq> []" have "(SOME q. distinct q \<and> set q = set rest) \<noteq> []" proof(rule someI2) from wq_distinct[OF vt, of cs] eq_wq show "distinct rest \<and> set rest = set rest" by auto next from ne show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> x \<noteq> []" by auto qed hence "hd (SOME q. distinct q \<and> set q = set rest) \<in> set (SOME q. distinct q \<and> set q = set rest)" by auto moreover have "set (SOME q. distinct q \<and> set q = set rest) = set rest" proof(rule someI2) from wq_distinct[OF vt, of cs] eq_wq show "distinct rest \<and> set rest = set rest" by auto next show "\<And>x. distinct x \<and> set x = set rest \<Longrightarrow> set x = set rest" by auto qed ultimately have "hd (SOME q. distinct q \<and> set q = set rest) \<in> set rest" by simp with eq_wq and wq_distinct[OF vt, of cs] show False by auto qedqed-- {* A useless definition *}definition cps:: "state \<Rightarrow> (thread \<times> precedence) set"where "cps s = {(th, cp s th) | th . th \<in> threads s}"text {* (* ddd *) One beauty of our modelling is that we follow the definitional extension tradition of HOL. The benefit of such a concise and miniature model is that large number of intuitively obvious facts are derived as lemmas, rather than asserted as axioms.*}text {* However, the lemmas in the forthcoming several locales are no longer obvious. These lemmas show how the current precedences should be recalculated after every execution step (in our model, every step is represented by an event, which in turn, represents a system call, or operation). Each operation is treated in a separate locale. The complication of current precedence recalculation comes because the changing of RAG needs to be taken into account, in addition to the changing of precedence. The reason RAG changing affects current precedence is that, according to the definition, current precedence of a thread is the maximum of the precedences of its dependants, where the dependants are defined in terms of RAG. Therefore, each operation, lemmas concerning the change of the precedences and RAG are derived first, so that the lemmas about current precedence recalculation can be based on.*}text {* (* ddd *) The following locale @{text "step_set_cps"} investigates the recalculation after the @{text "Set"} operation.*}locale step_set_cps = fixes s' th prio s -- {* @{text "s'"} is the system state before the operation *} -- {* @{text "s"} is the system state after the operation *} defines s_def : "s \<equiv> (Set th prio#s')" -- {* @{text "s"} is assumed to be a legitimate state, from which the legitimacy of @{text "s"} can be derived. *} assumes vt_s: "vt s"context step_set_cps begininterpretation rtree_RAGs: rtree "RAG s"proof show "single_valued (RAG s)" apply (intro_locales) by (unfold single_valued_def, auto intro: unique_RAG[OF vt_s]) show "acyclic (RAG s)" by (rule acyclic_RAG[OF vt_s])qedinterpretation rtree_RAGs': rtree "RAG s'"proof show "single_valued (RAG s')" apply (intro_locales) by (unfold single_valued_def, auto intro: unique_RAG[OF step_back_vt[OF vt_s[unfolded s_def]]]) show "acyclic (RAG s')" by (rule acyclic_RAG[OF step_back_vt[OF vt_s[unfolded s_def]]])qedtext {* (* ddd *) The following lemma confirms that @{text "Set"}-operating only changes the precedence of initiating thread.*}lemma eq_preced: fixes th' assumes "th' \<noteq> th" shows "preced th' s = preced th' s'"proof - from assms show ?thesis by (unfold s_def, auto simp:preced_def)qedtext {* (* ddd *) The following lemma assures that the resetting of priority does not change the RAG. *}lemma eq_dep: "RAG s = RAG s'" by (unfold s_def RAG_set_unchanged, auto)text {* Th following lemma @{text "eq_cp_pre"} circumscribe a rough range of recalculation. It says, thread other than the initiating thread @{text "th"} does not need recalculation unless it lies upstream of @{text "th"} in the RAG. The reason behind this lemma is that: the change of precedence of one thread can only affect it's upstream threads, according to lemma @{text "eq_preced"}. Since the only thread which might change precedence is @{text "th"}, so only @{text "th"} and its upstream threads need recalculation. (* ccc *)*}lemma eq_cp_pre: fixes th' assumes neq_th: "th' \<noteq> th" and nd: "th \<notin> dependants s th'" shows "cp s th' = cp s' th'"proof - -- {* This is what we need to prove after expanding the definition of @{text "cp"} *} have "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')) = Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependants (wq s') th'))" (is "Max (?f1 ` ({th'} \<union> ?A)) = Max (?f2 ` ({th'} \<union> ?B))") proof - -- {* Since RAG is not changed by @{text "Set"}-operation, the dependants of any threads are not changed, this is one of key facts underpinning this lemma *} have eq_ab: "?A = ?B" by (unfold cs_dependants_def, auto simp:eq_dep eq_RAG) have "(?f1 ` ({th'} \<union> ?A)) = (?f2 ` ({th'} \<union> ?B))" proof(rule image_cong) show "{th'} \<union> ?A = {th'} \<union> ?B" by (simp only:eq_ab) next fix x assume x_in: "x \<in> {th'} \<union> ?B" show "?f1 x = ?f2 x" proof(rule eq_preced) -- {* The other key fact underpinning this lemma is @{text "eq_preced"} *} from x_in[folded eq_ab, unfolded eq_dependants] have "x \<in> {th'} \<union> dependants s th'" . thus "x \<noteq> th" proof assume "x \<in> {th'}" with `th' \<noteq> th` show ?thesis by simp next assume "x \<in> dependants s th'" with `th \<notin> dependants s th'` show ?thesis by auto qed qed qed thus ?thesis by simp qed thus ?thesis by (unfold cp_eq_cpreced cpreced_def)qedtext {* The following lemma shows that no thread lies upstream of the initiating thread @{text "th"}. The reason for this is that only no-blocked thread can initiate a system call. Since thread @{text "th"} is non-blocked, it is not waiting for any critical resource. Therefore, there is edge leading out of @{text "th"} in the RAG. Consequently, there is no node (neither resource nor thread) upstream of @{text "th"}.*}lemma no_dependants: shows "th \<notin> dependants s th'"proof assume "th \<in> dependants s th'" from `th \<in> dependants s th'` have "(Th th, Th th') \<in> (RAG s')\<^sup>+" by (unfold s_dependants_def, unfold eq_RAG, unfold eq_dep, auto) from tranclD[OF this] obtain z where "(Th th, z) \<in> RAG s'" by auto moreover have "th \<in> readys s'" proof - from step_back_step [OF vt_s[unfolded s_def]] have "step s' (Set th prio)" . hence "th \<in> runing s'" by (cases, simp) thus ?thesis by (simp add:readys_def runing_def) qed ultimately show "False" apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def) by (fold wq_def, blast)qed(* Result improved *)text {* A simple combination of @{text "eq_cp_pre"} and @{text "no_dependants"} gives the main lemma of this locale, which shows that only the initiating thread needs a recalculation of current precedence.*}lemma eq_cp: fixes th' assumes "th' \<noteq> th" shows "cp s th' = cp s' th'" by (rule eq_cp_pre[OF assms no_dependants])endtext {* The following @{text "step_v_cps"} is the locale for @{text "V"}-operation.*}locale step_v_cps = -- {* @{text "th"} is the initiating thread *} -- {* @{text "cs"} is the critical resource release by the @{text "V"}-operation *} fixes s' th cs s -- {* @{text "s'"} is the state before operation*} defines s_def : "s \<equiv> (V th cs#s')" -- {* @{text "s"} is the state after operation*} -- {* @{text "s"} is assumed to be valid, which implies the validity of @{text "s'"} *} assumes vt_s: "vt s"text {* The following @{text "step_v_cps_nt"} is the sub-locale for @{text "V"}-operation, which represents the case when there is another thread @{text "th'"} to take over the critical resource released by the initiating thread @{text "th"}.*}locale step_v_cps_nt = step_v_cps + fixes th' -- {* @{text "th'"} is assumed to take over @{text "cs"} *} assumes nt: "next_th s' th cs th'" context step_v_cps_ntbegintext {* Lemma @{text "RAG_s"} confirms the change of RAG: two edges removed and one added, as shown by the following diagram.*}(* RAG before the V-operation th1 ----| | th' ----| |----> cs -----| th2 ----| | | | th3 ----| | |------> th th4 ----| | | | th5 ----| | |----> cs'-----| th6 ----| | th7 ----| RAG after the V-operation th1 ----| | |----> cs ----> th' th2 ----| | th3 ----| th4 ----| | th5 ----| |----> cs'----> th th6 ----| | th7 ----|*)lemma RAG_s: "RAG s = (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) \<union> {(Cs cs, Th th')}"proof - from step_RAG_v[OF vt_s[unfolded s_def], folded s_def] and nt show ?thesis by (auto intro:next_th_unique)qedtext {* Lemma @{text "dependants_kept"} shows only @{text "th"} and @{text "th'"} have their dependants changed.*}lemma dependants_kept: fixes th'' assumes neq1: "th'' \<noteq> th" and neq2: "th'' \<noteq> th'" shows "dependants (wq s) th'' = dependants (wq s') th''"proof(auto) (* ccc *) fix x assume "x \<in> dependants (wq s) th''" hence dp: "(Th x, Th th'') \<in> (RAG s)^+" by (auto simp:cs_dependants_def eq_RAG) { fix n have "(n, Th th'') \<in> (RAG s)^+ \<Longrightarrow> (n, Th th'') \<in> (RAG s')^+" proof(induct rule:converse_trancl_induct) fix y assume "(y, Th th'') \<in> RAG s" with RAG_s neq1 neq2 have "(y, Th th'') \<in> RAG s'" by auto thus "(y, Th th'') \<in> (RAG s')\<^sup>+" by auto next fix y z assume yz: "(y, z) \<in> RAG s" and ztp: "(z, Th th'') \<in> (RAG s)\<^sup>+" and ztp': "(z, Th th'') \<in> (RAG s')\<^sup>+" have "y \<noteq> Cs cs \<and> y \<noteq> Th th'" proof show "y \<noteq> Cs cs" proof assume eq_y: "y = Cs cs" with yz have dp_yz: "(Cs cs, z) \<in> RAG s" by simp from RAG_s have cst': "(Cs cs, Th th') \<in> RAG s" by simp from unique_RAG[OF vt_s this dp_yz] have eq_z: "z = Th th'" by simp with ztp have "(Th th', Th th'') \<in> (RAG s)^+" by simp from converse_tranclE[OF this] obtain cs' where dp'': "(Th th', Cs cs') \<in> RAG s" by (auto simp:s_RAG_def) with RAG_s have dp': "(Th th', Cs cs') \<in> RAG s'" by auto from dp'' eq_y yz eq_z have "(Cs cs, Cs cs') \<in> (RAG s)^+" by auto moreover have "cs' = cs" proof - from next_waiting[OF step_back_vt[OF vt_s[unfolded s_def]] nt] have "(Th th', Cs cs) \<in> RAG s'" by (auto simp:s_waiting_def wq_def s_RAG_def cs_waiting_def) from unique_RAG[OF step_back_vt[OF vt_s[unfolded s_def]] this dp'] show ?thesis by simp qed ultimately have "(Cs cs, Cs cs) \<in> (RAG s)^+" by simp moreover note wf_trancl[OF wf_RAG[OF vt_s]] ultimately show False by auto qed next show "y \<noteq> Th th'" proof assume eq_y: "y = Th th'" with yz have dps: "(Th th', z) \<in> RAG s" by simp with RAG_s have dps': "(Th th', z) \<in> RAG s'" by auto have "z = Cs cs" proof - from next_waiting[OF step_back_vt[OF vt_s[unfolded s_def]] nt] have "(Th th', Cs cs) \<in> RAG s'" by (auto simp:s_waiting_def wq_def s_RAG_def cs_waiting_def) from unique_RAG[OF step_back_vt[OF vt_s[unfolded s_def]] dps' this] show ?thesis . qed with dps RAG_s show False by auto qed qed with RAG_s yz have "(y, z) \<in> RAG s'" by auto with ztp' show "(y, Th th'') \<in> (RAG s')\<^sup>+" by auto qed } from this[OF dp] show "x \<in> dependants (wq s') th''" by (auto simp:cs_dependants_def eq_RAG)next fix x assume "x \<in> dependants (wq s') th''" hence dp: "(Th x, Th th'') \<in> (RAG s')^+" by (auto simp:cs_dependants_def eq_RAG) { fix n have "(n, Th th'') \<in> (RAG s')^+ \<Longrightarrow> (n, Th th'') \<in> (RAG s)^+" proof(induct rule:converse_trancl_induct) fix y assume "(y, Th th'') \<in> RAG s'" with RAG_s neq1 neq2 have "(y, Th th'') \<in> RAG s" by auto thus "(y, Th th'') \<in> (RAG s)\<^sup>+" by auto next fix y z assume yz: "(y, z) \<in> RAG s'" and ztp: "(z, Th th'') \<in> (RAG s')\<^sup>+" and ztp': "(z, Th th'') \<in> (RAG s)\<^sup>+" have "y \<noteq> Cs cs \<and> y \<noteq> Th th'" proof show "y \<noteq> Cs cs" proof assume eq_y: "y = Cs cs" with yz have dp_yz: "(Cs cs, z) \<in> RAG s'" by simp from this have eq_z: "z = Th th" proof - from step_back_step[OF vt_s[unfolded s_def]] have "(Cs cs, Th th) \<in> RAG s'" by(cases, auto simp: wq_def s_RAG_def cs_holding_def s_holding_def) from unique_RAG[OF step_back_vt[OF vt_s[unfolded s_def]] this dp_yz] show ?thesis by simp qed from converse_tranclE[OF ztp] obtain u where "(z, u) \<in> RAG s'" by auto moreover from step_back_step[OF vt_s[unfolded s_def]] have "th \<in> readys s'" by (cases, simp add:runing_def) moreover note eq_z ultimately show False by (auto simp:readys_def wq_def s_RAG_def s_waiting_def cs_waiting_def) qed next show "y \<noteq> Th th'" proof assume eq_y: "y = Th th'" with yz have dps: "(Th th', z) \<in> RAG s'" by simp have "z = Cs cs" proof - from next_waiting[OF step_back_vt[OF vt_s[unfolded s_def]] nt] have "(Th th', Cs cs) \<in> RAG s'" by (auto simp:s_waiting_def wq_def s_RAG_def cs_waiting_def) from unique_RAG[OF step_back_vt[OF vt_s[unfolded s_def]] dps this] show ?thesis . qed with ztp have cs_i: "(Cs cs, Th th'') \<in> (RAG s')\<^sup>+" by simp from step_back_step[OF vt_s[unfolded s_def]] have cs_th: "(Cs cs, Th th) \<in> RAG s'" by(cases, auto simp: s_RAG_def wq_def cs_holding_def s_holding_def) have "(Cs cs, Th th'') \<notin> RAG s'" proof assume "(Cs cs, Th th'') \<in> RAG s'" from unique_RAG[OF step_back_vt[OF vt_s[unfolded s_def]] this cs_th] and neq1 show "False" by simp qed with converse_tranclE[OF cs_i] obtain u where cu: "(Cs cs, u) \<in> RAG s'" and u_t: "(u, Th th'') \<in> (RAG s')\<^sup>+" by auto have "u = Th th" proof - from unique_RAG[OF step_back_vt[OF vt_s[unfolded s_def]] cu cs_th] show ?thesis . qed with u_t have "(Th th, Th th'') \<in> (RAG s')\<^sup>+" by simp from converse_tranclE[OF this] obtain v where "(Th th, v) \<in> (RAG s')" by auto moreover from step_back_step[OF vt_s[unfolded s_def]] have "th \<in> readys s'" by (cases, simp add:runing_def) ultimately show False by (auto simp:readys_def wq_def s_RAG_def s_waiting_def cs_waiting_def) qed qed with RAG_s yz have "(y, z) \<in> RAG s" by auto with ztp' show "(y, Th th'') \<in> (RAG s)\<^sup>+" by auto qed } from this[OF dp] show "x \<in> dependants (wq s) th''" by (auto simp:cs_dependants_def eq_RAG)qedlemma cp_kept: fixes th'' assumes neq1: "th'' \<noteq> th" and neq2: "th'' \<noteq> th'" shows "cp s th'' = cp s' th''"proof - from dependants_kept[OF neq1 neq2] have "dependants (wq s) th'' = dependants (wq s') th''" . moreover { fix th1 assume "th1 \<in> dependants (wq s) th''" have "preced th1 s = preced th1 s'" by (unfold s_def, auto simp:preced_def) } moreover have "preced th'' s = preced th'' s'" by (unfold s_def, auto simp:preced_def) ultimately have "((\<lambda>th. preced th s) ` ({th''} \<union> dependants (wq s) th'')) = ((\<lambda>th. preced th s') ` ({th''} \<union> dependants (wq s') th''))" by (auto simp:image_def) thus ?thesis by (unfold cp_eq_cpreced cpreced_def, simp)qedendlocale step_v_cps_nnt = step_v_cps + assumes nnt: "\<And> th'. (\<not> next_th s' th cs th')"context step_v_cps_nntbeginlemma nw_cs: "(Th th1, Cs cs) \<notin> RAG s'"proof assume "(Th th1, Cs cs) \<in> RAG s'" thus "False" apply (auto simp:s_RAG_def cs_waiting_def) proof - assume h1: "th1 \<in> set (wq s' cs)" and h2: "th1 \<noteq> hd (wq s' cs)" from step_back_step[OF vt_s[unfolded s_def]] show "False" proof(cases) assume "holding s' th cs" then obtain rest where eq_wq: "wq s' cs = th#rest" apply (unfold s_holding_def wq_def[symmetric]) by (case_tac "(wq s' cs)", auto) with h1 h2 have ne: "rest \<noteq> []" by auto with eq_wq have "next_th s' th cs (hd (SOME q. distinct q \<and> set q = set rest))" by(unfold next_th_def, rule_tac x = "rest" in exI, auto) with nnt show ?thesis by auto qed qedqedlemma RAG_s: "RAG s = RAG s' - {(Cs cs, Th th)}"proof - from nnt and step_RAG_v[OF vt_s[unfolded s_def], folded s_def] show ?thesis by autoqedlemma child_kept_left: assumes "(n1, n2) \<in> (child s')^+" shows "(n1, n2) \<in> (child s)^+"proof - from assms show ?thesis proof(induct rule: converse_trancl_induct) case (base y) from base obtain th1 cs1 th2 where h1: "(Th th1, Cs cs1) \<in> RAG s'" and h2: "(Cs cs1, Th th2) \<in> RAG s'" and eq_y: "y = Th th1" and eq_n2: "n2 = Th th2" by (auto simp:child_def) have "cs1 \<noteq> cs" proof assume eq_cs: "cs1 = cs" with h1 have "(Th th1, Cs cs1) \<in> RAG s'" by simp with nw_cs eq_cs show False by auto qed with h1 h2 RAG_s have h1': "(Th th1, Cs cs1) \<in> RAG s" and h2': "(Cs cs1, Th th2) \<in> RAG s" by auto hence "(Th th1, Th th2) \<in> child s" by (auto simp:child_def) with eq_y eq_n2 have "(y, n2) \<in> child s" by simp thus ?case by auto next case (step y z) have "(y, z) \<in> child s'" by fact then obtain th1 cs1 th2 where h1: "(Th th1, Cs cs1) \<in> RAG s'" and h2: "(Cs cs1, Th th2) \<in> RAG s'" and eq_y: "y = Th th1" and eq_z: "z = Th th2" by (auto simp:child_def) have "cs1 \<noteq> cs" proof assume eq_cs: "cs1 = cs" with h1 have "(Th th1, Cs cs1) \<in> RAG s'" by simp with nw_cs eq_cs show False by auto qed with h1 h2 RAG_s have h1': "(Th th1, Cs cs1) \<in> RAG s" and h2': "(Cs cs1, Th th2) \<in> RAG s" by auto hence "(Th th1, Th th2) \<in> child s" by (auto simp:child_def) with eq_y eq_z have "(y, z) \<in> child s" by simp moreover have "(z, n2) \<in> (child s)^+" by fact ultimately show ?case by auto qedqedlemma child_kept_right: assumes "(n1, n2) \<in> (child s)^+" shows "(n1, n2) \<in> (child s')^+"proof - from assms show ?thesis proof(induct) case (base y) from base and RAG_s have "(n1, y) \<in> child s'" by (auto simp:child_def) thus ?case by auto next case (step y z) have "(y, z) \<in> child s" by fact with RAG_s have "(y, z) \<in> child s'" by (auto simp:child_def) moreover have "(n1, y) \<in> (child s')\<^sup>+" by fact ultimately show ?case by auto qedqedlemma eq_child: "(child s)^+ = (child s')^+" by (insert child_kept_left child_kept_right, auto)lemma eq_cp: fixes th' shows "cp s th' = cp s' th'" apply (unfold cp_eq_cpreced cpreced_def)proof - have eq_dp: "\<And> th. dependants (wq s) th = dependants (wq s') th" apply (unfold cs_dependants_def, unfold eq_RAG) proof - from eq_child have "\<And>th. {th'. (Th th', Th th) \<in> (child s)\<^sup>+} = {th'. (Th th', Th th) \<in> (child s')\<^sup>+}" by simp with child_RAG_eq[OF vt_s] child_RAG_eq[OF step_back_vt[OF vt_s[unfolded s_def]]] show "\<And>th. {th'. (Th th', Th th) \<in> (RAG s)\<^sup>+} = {th'. (Th th', Th th) \<in> (RAG s')\<^sup>+}" by simp qed moreover { fix th1 assume "th1 \<in> {th'} \<union> dependants (wq s') th'" hence "th1 = th' \<or> th1 \<in> dependants (wq s') th'" by auto hence "preced th1 s = preced th1 s'" proof assume "th1 = th'" show "preced th1 s = preced th1 s'" by (simp add:s_def preced_def) next assume "th1 \<in> dependants (wq s') th'" show "preced th1 s = preced th1 s'" by (simp add:s_def preced_def) qed } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')) = ((\<lambda>th. preced th s') ` ({th'} \<union> dependants (wq s') th'))" by (auto simp:image_def) thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')) = Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependants (wq s') th'))" by simpqedendlocale step_P_cps = fixes s' th cs s defines s_def : "s \<equiv> (P th cs#s')" assumes vt_s: "vt s"locale step_P_cps_ne =step_P_cps + assumes ne: "wq s' cs \<noteq> []"locale step_P_cps_e =step_P_cps + assumes ee: "wq s' cs = []"context step_P_cps_ebeginlemma RAG_s: "RAG s = RAG s' \<union> {(Cs cs, Th th)}"proof - from ee and step_RAG_p[OF vt_s[unfolded s_def], folded s_def] show ?thesis by autoqedlemma child_kept_left: assumes "(n1, n2) \<in> (child s')^+" shows "(n1, n2) \<in> (child s)^+"proof - from assms show ?thesis proof(induct rule: converse_trancl_induct) case (base y) from base obtain th1 cs1 th2 where h1: "(Th th1, Cs cs1) \<in> RAG s'" and h2: "(Cs cs1, Th th2) \<in> RAG s'" and eq_y: "y = Th th1" and eq_n2: "n2 = Th th2" by (auto simp:child_def) have "cs1 \<noteq> cs" proof assume eq_cs: "cs1 = cs" with h1 have "(Th th1, Cs cs) \<in> RAG s'" by simp with ee show False by (auto simp:s_RAG_def cs_waiting_def) qed with h1 h2 RAG_s have h1': "(Th th1, Cs cs1) \<in> RAG s" and h2': "(Cs cs1, Th th2) \<in> RAG s" by auto hence "(Th th1, Th th2) \<in> child s" by (auto simp:child_def) with eq_y eq_n2 have "(y, n2) \<in> child s" by simp thus ?case by auto next case (step y z) have "(y, z) \<in> child s'" by fact then obtain th1 cs1 th2 where h1: "(Th th1, Cs cs1) \<in> RAG s'" and h2: "(Cs cs1, Th th2) \<in> RAG s'" and eq_y: "y = Th th1" and eq_z: "z = Th th2" by (auto simp:child_def) have "cs1 \<noteq> cs" proof assume eq_cs: "cs1 = cs" with h1 have "(Th th1, Cs cs) \<in> RAG s'" by simp with ee show False by (auto simp:s_RAG_def cs_waiting_def) qed with h1 h2 RAG_s have h1': "(Th th1, Cs cs1) \<in> RAG s" and h2': "(Cs cs1, Th th2) \<in> RAG s" by auto hence "(Th th1, Th th2) \<in> child s" by (auto simp:child_def) with eq_y eq_z have "(y, z) \<in> child s" by simp moreover have "(z, n2) \<in> (child s)^+" by fact ultimately show ?case by auto qedqedlemma child_kept_right: assumes "(n1, n2) \<in> (child s)^+" shows "(n1, n2) \<in> (child s')^+"proof - from assms show ?thesis proof(induct) case (base y) from base and RAG_s have "(n1, y) \<in> child s'" apply (auto simp:child_def) proof - fix th' assume "(Th th', Cs cs) \<in> RAG s'" with ee have "False" by (auto simp:s_RAG_def cs_waiting_def) thus "\<exists>cs. (Th th', Cs cs) \<in> RAG s' \<and> (Cs cs, Th th) \<in> RAG s'" by auto qed thus ?case by auto next case (step y z) have "(y, z) \<in> child s" by fact with RAG_s have "(y, z) \<in> child s'" apply (auto simp:child_def) proof - fix th' assume "(Th th', Cs cs) \<in> RAG s'" with ee have "False" by (auto simp:s_RAG_def cs_waiting_def) thus "\<exists>cs. (Th th', Cs cs) \<in> RAG s' \<and> (Cs cs, Th th) \<in> RAG s'" by auto qed moreover have "(n1, y) \<in> (child s')\<^sup>+" by fact ultimately show ?case by auto qedqedlemma eq_child: "(child s)^+ = (child s')^+" by (insert child_kept_left child_kept_right, auto)lemma eq_cp: fixes th' shows "cp s th' = cp s' th'" apply (unfold cp_eq_cpreced cpreced_def)proof - have eq_dp: "\<And> th. dependants (wq s) th = dependants (wq s') th" apply (unfold cs_dependants_def, unfold eq_RAG) proof - from eq_child have "\<And>th. {th'. (Th th', Th th) \<in> (child s)\<^sup>+} = {th'. (Th th', Th th) \<in> (child s')\<^sup>+}" by auto with child_RAG_eq[OF vt_s] child_RAG_eq[OF step_back_vt[OF vt_s[unfolded s_def]]] show "\<And>th. {th'. (Th th', Th th) \<in> (RAG s)\<^sup>+} = {th'. (Th th', Th th) \<in> (RAG s')\<^sup>+}" by simp qed moreover { fix th1 assume "th1 \<in> {th'} \<union> dependants (wq s') th'" hence "th1 = th' \<or> th1 \<in> dependants (wq s') th'" by auto hence "preced th1 s = preced th1 s'" proof assume "th1 = th'" show "preced th1 s = preced th1 s'" by (simp add:s_def preced_def) next assume "th1 \<in> dependants (wq s') th'" show "preced th1 s = preced th1 s'" by (simp add:s_def preced_def) qed } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')) = ((\<lambda>th. preced th s') ` ({th'} \<union> dependants (wq s') th'))" by (auto simp:image_def) thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')) = Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependants (wq s') th'))" by simpqedendcontext step_P_cps_nebeginlemma RAG_s: "RAG s = RAG s' \<union> {(Th th, Cs cs)}"proof - from step_RAG_p[OF vt_s[unfolded s_def]] and ne show ?thesis by (simp add:s_def)qedlemma eq_child_left: assumes nd: "(Th th, Th th') \<notin> (child s)^+" shows "(n1, Th th') \<in> (child s)^+ \<Longrightarrow> (n1, Th th') \<in> (child s')^+"proof(induct rule:converse_trancl_induct) case (base y) from base obtain th1 cs1 where h1: "(Th th1, Cs cs1) \<in> RAG s" and h2: "(Cs cs1, Th th') \<in> RAG s" and eq_y: "y = Th th1" by (auto simp:child_def) have "th1 \<noteq> th" proof assume "th1 = th" with base eq_y have "(Th th, Th th') \<in> child s" by simp with nd show False by auto qed with h1 h2 RAG_s have h1': "(Th th1, Cs cs1) \<in> RAG s'" and h2': "(Cs cs1, Th th') \<in> RAG s'" by auto with eq_y show ?case by (auto simp:child_def)next case (step y z) have yz: "(y, z) \<in> child s" by fact then obtain th1 cs1 th2 where h1: "(Th th1, Cs cs1) \<in> RAG s" and h2: "(Cs cs1, Th th2) \<in> RAG s" and eq_y: "y = Th th1" and eq_z: "z = Th th2" by (auto simp:child_def) have "th1 \<noteq> th" proof assume "th1 = th" with yz eq_y have "(Th th, z) \<in> child s" by simp moreover have "(z, Th th') \<in> (child s)^+" by fact ultimately have "(Th th, Th th') \<in> (child s)^+" by auto with nd show False by auto qed with h1 h2 RAG_s have h1': "(Th th1, Cs cs1) \<in> RAG s'" and h2': "(Cs cs1, Th th2) \<in> RAG s'" by auto with eq_y eq_z have "(y, z) \<in> child s'" by (auto simp:child_def) moreover have "(z, Th th') \<in> (child s')^+" by fact ultimately show ?case by autoqedlemma eq_child_right: shows "(n1, Th th') \<in> (child s')^+ \<Longrightarrow> (n1, Th th') \<in> (child s)^+"proof(induct rule:converse_trancl_induct) case (base y) with RAG_s show ?case by (auto simp:child_def)next case (step y z) have "(y, z) \<in> child s'" by fact with RAG_s have "(y, z) \<in> child s" by (auto simp:child_def) moreover have "(z, Th th') \<in> (child s)^+" by fact ultimately show ?case by autoqedlemma eq_child: assumes nd: "(Th th, Th th') \<notin> (child s)^+" shows "((n1, Th th') \<in> (child s)^+) = ((n1, Th th') \<in> (child s')^+)" by (insert eq_child_left[OF nd] eq_child_right, auto)lemma eq_cp: fixes th' assumes nd: "th \<notin> dependants s th'" shows "cp s th' = cp s' th'" apply (unfold cp_eq_cpreced cpreced_def)proof - have nd': "(Th th, Th th') \<notin> (child s)^+" proof assume "(Th th, Th th') \<in> (child s)\<^sup>+" with child_RAG_eq[OF vt_s] have "(Th th, Th th') \<in> (RAG s)\<^sup>+" by simp with nd show False by (simp add:s_dependants_def eq_RAG) qed have eq_dp: "dependants (wq s) th' = dependants (wq s') th'" proof(auto) fix x assume " x \<in> dependants (wq s) th'" thus "x \<in> dependants (wq s') th'" apply (auto simp:cs_dependants_def eq_RAG) proof - assume "(Th x, Th th') \<in> (RAG s)\<^sup>+" with child_RAG_eq[OF vt_s] have "(Th x, Th th') \<in> (child s)\<^sup>+" by simp with eq_child[OF nd'] have "(Th x, Th th') \<in> (child s')^+" by simp with child_RAG_eq[OF step_back_vt[OF vt_s[unfolded s_def]]] show "(Th x, Th th') \<in> (RAG s')\<^sup>+" by simp qed next fix x assume "x \<in> dependants (wq s') th'" thus "x \<in> dependants (wq s) th'" apply (auto simp:cs_dependants_def eq_RAG) proof - assume "(Th x, Th th') \<in> (RAG s')\<^sup>+" with child_RAG_eq[OF step_back_vt[OF vt_s[unfolded s_def]]] have "(Th x, Th th') \<in> (child s')\<^sup>+" by simp with eq_child[OF nd'] have "(Th x, Th th') \<in> (child s)^+" by simp with child_RAG_eq[OF vt_s] show "(Th x, Th th') \<in> (RAG s)\<^sup>+" by simp qed qed moreover { fix th1 have "preced th1 s = preced th1 s'" by (simp add:s_def preced_def) } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')) = ((\<lambda>th. preced th s') ` ({th'} \<union> dependants (wq s') th'))" by (auto simp:image_def) thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')) = Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependants (wq s') th'))" by simpqedlemma eq_up: fixes th' th'' assumes dp1: "th \<in> dependants s th'" and dp2: "th' \<in> dependants s th''" and eq_cps: "cp s th' = cp s' th'" shows "cp s th'' = cp s' th''"proof - from dp2 have "(Th th', Th th'') \<in> (RAG (wq s))\<^sup>+" by (simp add:s_dependants_def) from RAG_child[OF vt_s this[unfolded eq_RAG]] have ch_th': "(Th th', Th th'') \<in> (child s)\<^sup>+" . moreover { fix n th'' have "\<lbrakk>(Th th', n) \<in> (child s)^+\<rbrakk> \<Longrightarrow> (\<forall> th'' . n = Th th'' \<longrightarrow> cp s th'' = cp s' th'')" proof(erule trancl_induct, auto) fix y th'' assume y_ch: "(y, Th th'') \<in> child s" and ih: "\<forall>th''. y = Th th'' \<longrightarrow> cp s th'' = cp s' th''" and ch': "(Th th', y) \<in> (child s)\<^sup>+" from y_ch obtain thy where eq_y: "y = Th thy" by (auto simp:child_def) with ih have eq_cpy:"cp s thy = cp s' thy" by blast from dp1 have "(Th th, Th th') \<in> (RAG s)^+" by (auto simp:s_dependants_def eq_RAG) moreover from child_RAG_p[OF ch'] and eq_y have "(Th th', Th thy) \<in> (RAG s)^+" by simp ultimately have dp_thy: "(Th th, Th thy) \<in> (RAG s)^+" by auto show "cp s th'' = cp s' th''" apply (subst cp_rec[OF vt_s]) proof - have "preced th'' s = preced th'' s'" by (simp add:s_def preced_def) moreover { fix th1 assume th1_in: "th1 \<in> children s th''" have "cp s th1 = cp s' th1" proof(cases "th1 = thy") case True with eq_cpy show ?thesis by simp next case False have neq_th1: "th1 \<noteq> th" proof assume eq_th1: "th1 = th" with dp_thy have "(Th th1, Th thy) \<in> (RAG s)^+" by simp from children_no_dep[OF vt_s _ _ this] and th1_in y_ch eq_y show False by (auto simp:children_def) qed have "th \<notin> dependants s th1" proof assume h:"th \<in> dependants s th1" from eq_y dp_thy have "th \<in> dependants s thy" by (auto simp:s_dependants_def eq_RAG) from dependants_child_unique[OF vt_s _ _ h this] th1_in y_ch eq_y have "th1 = thy" by (auto simp:children_def child_def) with False show False by auto qed from eq_cp[OF this] show ?thesis . qed } ultimately have "{preced th'' s} \<union> (cp s ` children s th'') = {preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def) moreover have "children s th'' = children s' th''" apply (unfold children_def child_def s_def RAG_set_unchanged, simp) apply (fold s_def, auto simp:RAG_s) proof - assume "(Cs cs, Th th'') \<in> RAG s'" with RAG_s have cs_th': "(Cs cs, Th th'') \<in> RAG s" by auto from dp1 have "(Th th, Th th') \<in> (RAG s)^+" by (auto simp:s_dependants_def eq_RAG) from converse_tranclE[OF this] obtain cs1 where h1: "(Th th, Cs cs1) \<in> RAG s" and h2: "(Cs cs1 , Th th') \<in> (RAG s)\<^sup>+" by (auto simp:s_RAG_def) have eq_cs: "cs1 = cs" proof - from RAG_s have "(Th th, Cs cs) \<in> RAG s" by simp from unique_RAG[OF vt_s this h1] show ?thesis by simp qed have False proof(rule converse_tranclE[OF h2]) assume "(Cs cs1, Th th') \<in> RAG s" with eq_cs have "(Cs cs, Th th') \<in> RAG s" by simp from unique_RAG[OF vt_s this cs_th'] have "th' = th''" by simp with ch' y_ch have "(Th th'', Th th'') \<in> (child s)^+" by auto with wf_trancl[OF wf_child[OF vt_s]] show False by auto next fix y assume "(Cs cs1, y) \<in> RAG s" and ytd: " (y, Th th') \<in> (RAG s)\<^sup>+" with eq_cs have csy: "(Cs cs, y) \<in> RAG s" by simp from unique_RAG[OF vt_s this cs_th'] have "y = Th th''" . with ytd have "(Th th'', Th th') \<in> (RAG s)^+" by simp from RAG_child[OF vt_s this] have "(Th th'', Th th') \<in> (child s)\<^sup>+" . moreover from ch' y_ch have ch'': "(Th th', Th th'') \<in> (child s)^+" by auto ultimately have "(Th th'', Th th'') \<in> (child s)^+" by auto with wf_trancl[OF wf_child[OF vt_s]] show False by auto qed thus "\<exists>cs. (Th th, Cs cs) \<in> RAG s' \<and> (Cs cs, Th th'') \<in> RAG s'" by auto qed ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''" by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]]) qed next fix th'' assume dp': "(Th th', Th th'') \<in> child s" show "cp s th'' = cp s' th''" apply (subst cp_rec[OF vt_s]) proof - have "preced th'' s = preced th'' s'" by (simp add:s_def preced_def) moreover { fix th1 assume th1_in: "th1 \<in> children s th''" have "cp s th1 = cp s' th1" proof(cases "th1 = th'") case True with eq_cps show ?thesis by simp next case False have neq_th1: "th1 \<noteq> th" proof assume eq_th1: "th1 = th" with dp1 have "(Th th1, Th th') \<in> (RAG s)^+" by (auto simp:s_dependants_def eq_RAG) from children_no_dep[OF vt_s _ _ this] th1_in dp' show False by (auto simp:children_def) qed show ?thesis proof(rule eq_cp) show "th \<notin> dependants s th1" proof assume "th \<in> dependants s th1" from dependants_child_unique[OF vt_s _ _ this dp1] th1_in dp' have "th1 = th'" by (auto simp:children_def) with False show False by auto qed qed qed } ultimately have "{preced th'' s} \<union> (cp s ` children s th'') = {preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def) moreover have "children s th'' = children s' th''" apply (unfold children_def child_def s_def RAG_set_unchanged, simp) apply (fold s_def, auto simp:RAG_s) proof - assume "(Cs cs, Th th'') \<in> RAG s'" with RAG_s have cs_th': "(Cs cs, Th th'') \<in> RAG s" by auto from dp1 have "(Th th, Th th') \<in> (RAG s)^+" by (auto simp:s_dependants_def eq_RAG) from converse_tranclE[OF this] obtain cs1 where h1: "(Th th, Cs cs1) \<in> RAG s" and h2: "(Cs cs1 , Th th') \<in> (RAG s)\<^sup>+" by (auto simp:s_RAG_def) have eq_cs: "cs1 = cs" proof - from RAG_s have "(Th th, Cs cs) \<in> RAG s" by simp from unique_RAG[OF vt_s this h1] show ?thesis by simp qed have False proof(rule converse_tranclE[OF h2]) assume "(Cs cs1, Th th') \<in> RAG s" with eq_cs have "(Cs cs, Th th') \<in> RAG s" by simp from unique_RAG[OF vt_s this cs_th'] have "th' = th''" by simp with dp' have "(Th th'', Th th'') \<in> (child s)^+" by auto with wf_trancl[OF wf_child[OF vt_s]] show False by auto next fix y assume "(Cs cs1, y) \<in> RAG s" and ytd: " (y, Th th') \<in> (RAG s)\<^sup>+" with eq_cs have csy: "(Cs cs, y) \<in> RAG s" by simp from unique_RAG[OF vt_s this cs_th'] have "y = Th th''" . with ytd have "(Th th'', Th th') \<in> (RAG s)^+" by simp from RAG_child[OF vt_s this] have "(Th th'', Th th') \<in> (child s)\<^sup>+" . moreover from dp' have ch'': "(Th th', Th th'') \<in> (child s)^+" by auto ultimately have "(Th th'', Th th'') \<in> (child s)^+" by auto with wf_trancl[OF wf_child[OF vt_s]] show False by auto qed thus "\<exists>cs. (Th th, Cs cs) \<in> RAG s' \<and> (Cs cs, Th th'') \<in> RAG s'" by auto qed ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''" by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]]) qed qed } ultimately show ?thesis by autoqedendlocale step_create_cps = fixes s' th prio s defines s_def : "s \<equiv> (Create th prio#s')" assumes vt_s: "vt s"context step_create_cpsbeginlemma eq_dep: "RAG s = RAG s'" by (unfold s_def RAG_create_unchanged, auto)lemma eq_cp: fixes th' assumes neq_th: "th' \<noteq> th" shows "cp s th' = cp s' th'" apply (unfold cp_eq_cpreced cpreced_def)proof - have nd: "th \<notin> dependants s th'" proof assume "th \<in> dependants s th'" hence "(Th th, Th th') \<in> (RAG s)^+" by (simp add:s_dependants_def eq_RAG) with eq_dep have "(Th th, Th th') \<in> (RAG s')^+" by simp from converse_tranclE[OF this] obtain y where "(Th th, y) \<in> RAG s'" by auto with dm_RAG_threads[OF step_back_vt[OF vt_s[unfolded s_def]]] have in_th: "th \<in> threads s'" by auto from step_back_step[OF vt_s[unfolded s_def]] show False proof(cases) assume "th \<notin> threads s'" with in_th show ?thesis by simp qed qed have eq_dp: "\<And> th. dependants (wq s) th = dependants (wq s') th" by (unfold cs_dependants_def, auto simp:eq_dep eq_RAG) moreover { fix th1 assume "th1 \<in> {th'} \<union> dependants (wq s') th'" hence "th1 = th' \<or> th1 \<in> dependants (wq s') th'" by auto hence "preced th1 s = preced th1 s'" proof assume "th1 = th'" with neq_th show "preced th1 s = preced th1 s'" by (auto simp:s_def preced_def) next assume "th1 \<in> dependants (wq s') th'" with nd and eq_dp have "th1 \<noteq> th" by (auto simp:eq_RAG cs_dependants_def s_dependants_def eq_dep) thus "preced th1 s = preced th1 s'" by (auto simp:s_def preced_def) qed } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')) = ((\<lambda>th. preced th s') ` ({th'} \<union> dependants (wq s') th'))" by (auto simp:image_def) thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')) = Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependants (wq s') th'))" by simpqedlemma nil_dependants: "dependants s th = {}"proof - from step_back_step[OF vt_s[unfolded s_def]] show ?thesis proof(cases) assume "th \<notin> threads s'" from not_thread_holdents[OF step_back_vt[OF vt_s[unfolded s_def]] this] have hdn: " holdents s' th = {}" . have "dependants s' th = {}" proof - { assume "dependants s' th \<noteq> {}" then obtain th' where dp: "(Th th', Th th) \<in> (RAG s')^+" by (auto simp:s_dependants_def eq_RAG) from tranclE[OF this] obtain cs' where "(Cs cs', Th th) \<in> RAG s'" by (auto simp:s_RAG_def) with hdn have False by (auto simp:holdents_test) } thus ?thesis by auto qed thus ?thesis by (unfold s_def s_dependants_def eq_RAG RAG_create_unchanged, simp) qedqedlemma eq_cp_th: "cp s th = preced th s" apply (unfold cp_eq_cpreced cpreced_def) by (insert nil_dependants, unfold s_dependants_def cs_dependants_def, auto)endlocale step_exit_cps = fixes s' th prio s defines s_def : "s \<equiv> Exit th # s'" assumes vt_s: "vt s"context step_exit_cpsbeginlemma eq_dep: "RAG s = RAG s'" by (unfold s_def RAG_exit_unchanged, auto)lemma eq_cp: fixes th' assumes neq_th: "th' \<noteq> th" shows "cp s th' = cp s' th'" apply (unfold cp_eq_cpreced cpreced_def)proof - have nd: "th \<notin> dependants s th'" proof assume "th \<in> dependants s th'" hence "(Th th, Th th') \<in> (RAG s)^+" by (simp add:s_dependants_def eq_RAG) with eq_dep have "(Th th, Th th') \<in> (RAG s')^+" by simp from converse_tranclE[OF this] obtain cs' where bk: "(Th th, Cs cs') \<in> RAG s'" by (auto simp:s_RAG_def) from step_back_step[OF vt_s[unfolded s_def]] show False proof(cases) assume "th \<in> runing s'" with bk show ?thesis apply (unfold runing_def readys_def s_waiting_def s_RAG_def) by (auto simp:cs_waiting_def wq_def) qed qed have eq_dp: "\<And> th. dependants (wq s) th = dependants (wq s') th" by (unfold cs_dependants_def, auto simp:eq_dep eq_RAG) moreover { fix th1 assume "th1 \<in> {th'} \<union> dependants (wq s') th'" hence "th1 = th' \<or> th1 \<in> dependants (wq s') th'" by auto hence "preced th1 s = preced th1 s'" proof assume "th1 = th'" with neq_th show "preced th1 s = preced th1 s'" by (auto simp:s_def preced_def) next assume "th1 \<in> dependants (wq s') th'" with nd and eq_dp have "th1 \<noteq> th" by (auto simp:eq_RAG cs_dependants_def s_dependants_def eq_dep) thus "preced th1 s = preced th1 s'" by (auto simp:s_def preced_def) qed } ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')) = ((\<lambda>th. preced th s') ` ({th'} \<union> dependants (wq s') th'))" by (auto simp:image_def) thus "Max ((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')) = Max ((\<lambda>th. preced th s') ` ({th'} \<union> dependants (wq s') th'))" by simpqedendend