pip: comparison Implementation.thy

equal deleted inserted replaced

-:0525670d8e6a
+:4763aa246dbd
 section {*
 This file contains lemmas used to guide the recalculation of current precedence
 after every system call (or system operation)
 *}
-theory Implementation
+theory ExtGG
-imports PIPBasics
+imports CpsG
 begin
 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
 recalculation are based on.
 *}
 section {* The @{term Set} operation *}
-text {* (* ddd *)
+context valid_trace_set
-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"
-sublocale step_set_cps < vat_s : valid_trace "s"
-proof
-from vt_s show "vt s" .
-qed
-sublocale step_set_cps < vat_s' : valid_trace "s'"
-proof
-from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
-qed
-context step_set_cps
 begin
 text {* (* ddd *)
 The following two lemmas confirm that @{text "Set"}-operation
 only changes the precedence of the initiating thread (or actor)
 of the operation (or event).
 *}
 lemma eq_preced:
 assumes "th' \<noteq> th"
-shows "preced th' s = preced th' s'"
+shows "preced th' (e#s) = preced th' s"
 proof -
 from assms show ?thesis
-by (unfold s_def, auto simp:preced_def)
+by (unfold is_set, auto simp:preced_def)
 qed
 lemma eq_the_preced:
 assumes "th' \<noteq> th"
-shows "the_preced s th' = the_preced s' th'"
+shows "the_preced (e#s) th' = the_preced s th'"
 using assms
 by (unfold the_preced_def, intro eq_preced, simp)
-text {*
-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 {* (* ddd *)
 Th following lemma @{text "eq_cp_pre"} says that the priority change of @{text "th"}
 only affects those threads, which as @{text "Th th"} in their sub-trees.
 The proof of this lemma is simplified by using the alternative definition
 of @{text "cp"}.
 *}
 lemma eq_cp_pre:
-assumes nd: "Th th \<notin> subtree (RAG s') (Th th')"
+assumes nd: "Th th \<notin> subtree (RAG s) (Th th')"
-shows "cp s th' = cp s' th'"
+shows "cp (e#s) th' = cp s th'"
 proof -
 -- {* After unfolding using the alternative definition, elements
 affecting the @{term "cp"}-value of threads become explicit.
 We only need to prove the following: *}
-have "Max (the_preced s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')}) =
+have "Max (the_preced (e#s) ` {th'a. Th th'a \<in> subtree (RAG (e#s)) (Th th')}) =
-Max (the_preced s' ` {th'a. Th th'a \<in> subtree (RAG s') (Th th')})"
+Max (the_preced s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')})"
 (is "Max (?f ` ?S1) = Max (?g ` ?S2)")
 proof -
 -- {* The base sets are equal. *}
-have "?S1 = ?S2" using eq_dep by simp
+have "?S1 = ?S2" using RAG_unchanged by simp
 -- {* The function values on the base set are equal as well. *}
 moreover have "\<forall> e \<in> ?S2. ?f e = ?g e"
 proof
 fix th1
 assume "th1 \<in> ?S2"
 with nd have "th1 \<noteq> th" by (auto)
 from eq_the_preced[OF this]
-show "the_preced s th1 = the_preced s' th1" .
+show "the_preced (e#s) th1 = the_preced s th1" .
 qed
 -- {* Therefore, the image of the functions are equal. *}
 ultimately have "(?f ` ?S1) = (?g ` ?S2)" by (auto intro!:f_image_eq)
 thus ?thesis by simp
 qed
 The following lemma shows that @{term "th"} is not in the
 sub-tree of any other thread.
 *}
 lemma th_in_no_subtree:
 assumes "th' \<noteq> th"
-shows "Th th \<notin> subtree (RAG s') (Th th')"
+shows "Th th \<notin> subtree (RAG s) (Th th')"
 proof -
-have "th \<in> readys s'"
+from readys_in_no_subtree[OF th_ready_s assms(1)]
-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
-from vat_s'.readys_in_no_subtree[OF this assms(1)]
 show ?thesis by blast
 qed
 text {*
 By combining @{thm "eq_cp_pre"} and @{thm "th_in_no_subtree"},
 it is obvious that the change of priority only affects the @{text "cp"}-value
 of the initiating thread @{text "th"}.
 *}
 lemma eq_cp:
 assumes "th' \<noteq> th"
-shows "cp s th' = cp s' th'"
+shows "cp (e#s) th' = cp s th'"
 by (rule eq_cp_pre[OF th_in_no_subtree[OF assms]])
 end
 section {* The @{term V} operation *}
 text {*
 The following @{text "step_v_cps"} is the locale for @{text "V"}-operation.
 *}
-locale step_v_cps =
--- {* @{text "th"} is the initiating thread *}
+context valid_trace_v
--- {* @{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"
-sublocale step_v_cps < vat_s : valid_trace "s"
-proof
-from vt_s show "vt s" .
-qed
-sublocale step_v_cps < vat_s' : valid_trace "s'"
-proof
-from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
-qed
-context step_v_cps
 begin
-lemma ready_th_s': "th \<in> readys s'"
+lemma ancestors_th: "ancestors (RAG s) (Th th) = {}"
-using step_back_step[OF vt_s[unfolded s_def]]
+proof -
-by (cases, simp add:runing_def)
+from readys_root[OF th_ready_s]
-lemma ancestors_th: "ancestors (RAG s') (Th th) = {}"
-proof -
-from vat_s'.readys_root[OF ready_th_s']
 show ?thesis
 by (unfold root_def, simp)
 qed
-lemma holding_th: "holding s' th cs"
-proof -
-from vt_s[unfolded s_def]
-have " PIP s' (V th cs)" by (cases, simp)
-thus ?thesis by (cases, auto)
-qed
 lemma edge_of_th:
-"(Cs cs, Th th) \<in> RAG s'"
+"(Cs cs, Th th) \<in> RAG s"
 proof -
-from holding_th
+from holding_th_cs_s
 show ?thesis
 by (unfold s_RAG_def holding_eq, auto)
 qed
 lemma ancestors_cs:
-"ancestors (RAG s') (Cs cs) = {Th th}"
+"ancestors (RAG s) (Cs cs) = {Th th}"
 proof -
-have "ancestors (RAG s') (Cs cs) = ancestors (RAG s') (Th th)  \<union>  {Th th}"
+have "ancestors (RAG s) (Cs cs) = ancestors (RAG s) (Th th)  \<union>  {Th th}"
-proof(rule vat_s'.rtree_RAG.ancestors_accum)
+by (rule rtree_RAG.ancestors_accum[OF edge_of_th])
-from vt_s[unfolded s_def]
-have " PIP s' (V th cs)" by (cases, simp)
-thus "(Cs cs, Th th) \<in> RAG s'"
-proof(cases)
-assume "holding s' th cs"
-from this[unfolded holding_eq]
-show ?thesis by (unfold s_RAG_def, auto)
-qed
-qed
 from this[unfolded ancestors_th] show ?thesis by simp
 qed
-lemma preced_kept: "the_preced s = the_preced s'"
-by (auto simp: s_def the_preced_def preced_def)
 end
 text {*
 The following @{text "step_v_cps_nt"} is the sub-locale for @{text "V"}-operation,
 which represents the case when there is another thread @{text "th'"}
 to take over the critical resource released by the initiating thread @{text "th"}.
 *}
-locale step_v_cps_nt = step_v_cps +
-fixes th'
+context valid_trace_v_n
--- {* @{text "th'"} is assumed to take over @{text "cs"} *}
-assumes nt: "next_th s' th cs th'"
-context step_v_cps_nt
 begin
-text {*
+lemma sub_RAGs':
-Lemma @{text "RAG_s"} confirms the change of RAG:
+"{(Cs cs, Th th), (Th taker, Cs cs)} \<subseteq> RAG s"
-two edges removed and one added, as shown by the following diagram.
+using next_th_RAG[OF next_th_taker]  .
-*}
-(*
-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 sub_RAGs': "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s'"
-using next_th_RAG[OF nt]  .
 lemma ancestors_th':
-"ancestors (RAG s') (Th th') = {Th th, Cs cs}"
+"ancestors (RAG s) (Th taker) = {Th th, Cs cs}"
 proof -
-have "ancestors (RAG s') (Th th') = ancestors (RAG s') (Cs cs) \<union> {Cs cs}"
+have "ancestors (RAG s) (Th taker) = ancestors (RAG s) (Cs cs) \<union> {Cs cs}"
-proof(rule  vat_s'.rtree_RAG.ancestors_accum)
+proof(rule  rtree_RAG.ancestors_accum)
-from sub_RAGs' show "(Th th', Cs cs) \<in> RAG s'" by auto
+from sub_RAGs' show "(Th taker, Cs cs) \<in> RAG s" by auto
 qed
 thus ?thesis using ancestors_th ancestors_cs by auto
 qed
 lemma RAG_s:
-"RAG s = (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) \<union>
+"RAG (e#s) = (RAG s - {(Cs cs, Th th), (Th taker, Cs cs)}) \<union>
-{(Cs cs, Th th')}"
+{(Cs cs, Th taker)}"
-proof -
+by (unfold RAG_es waiting_set_eq holding_set_eq, auto)
-from step_RAG_v[OF vt_s[unfolded s_def], folded s_def]
-and nt show ?thesis  by (auto intro:next_th_unique)
-qed
 lemma subtree_kept: (* ddd *)
-assumes "th1 \<notin> {th, th'}"
+assumes "th1 \<notin> {th, taker}"
-shows "subtree (RAG s) (Th th1) = subtree (RAG s') (Th th1)" (is "_ = ?R")
+shows "subtree (RAG (e#s)) (Th th1) =
-proof -
+subtree (RAG s) (Th th1)" (is "_ = ?R")
-let ?RAG' = "(RAG s' - {(Cs cs, Th th), (Th th', Cs cs)})"
+proof -
-let ?RAG'' = "?RAG' \<union> {(Cs cs, Th th')}"
+let ?RAG' = "(RAG s - {(Cs cs, Th th), (Th taker, Cs cs)})"
+let ?RAG'' = "?RAG' \<union> {(Cs cs, Th taker)}"
 have "subtree ?RAG' (Th th1) = ?R"
 proof(rule subset_del_subtree_outside)
-show "Range {(Cs cs, Th th), (Th th', Cs cs)} \<inter> subtree (RAG s') (Th th1) = {}"
+show "Range {(Cs cs, Th th), (Th taker, Cs cs)} \<inter> subtree (RAG s) (Th th1) = {}"
 proof -
-have "(Th th) \<notin> subtree (RAG s') (Th th1)"
+have "(Th th) \<notin> subtree (RAG s) (Th th1)"
 proof(rule subtree_refute)
-show "Th th1 \<notin> ancestors (RAG s') (Th th)"
+show "Th th1 \<notin> ancestors (RAG s) (Th th)"
 by (unfold ancestors_th, simp)
 next
 from assms show "Th th1 \<noteq> Th th" by simp
 qed
-moreover have "(Cs cs) \<notin>  subtree (RAG s') (Th th1)"
+moreover have "(Cs cs) \<notin>  subtree (RAG s) (Th th1)"
 proof(rule subtree_refute)
-show "Th th1 \<notin> ancestors (RAG s') (Cs cs)"
+show "Th th1 \<notin> ancestors (RAG s) (Cs cs)"
 by (unfold ancestors_cs, insert assms, auto)
 qed simp
-ultimately have "{Th th, Cs cs} \<inter> subtree (RAG s') (Th th1) = {}" by auto
+ultimately have "{Th th, Cs cs} \<inter> subtree (RAG s) (Th th1) = {}" by auto
 thus ?thesis by simp
 qed
 qed
 moreover have "subtree ?RAG'' (Th th1) =  subtree ?RAG' (Th th1)"
 proof(rule subtree_insert_next)
-show "Th th' \<notin> subtree (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) (Th th1)"
+show "Th taker \<notin> subtree (RAG s - {(Cs cs, Th th), (Th taker, Cs cs)}) (Th th1)"
 proof(rule subtree_refute)
-show "Th th1 \<notin> ancestors (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) (Th th')"
+show "Th th1 \<notin> ancestors (RAG s - {(Cs cs, Th th), (Th taker, Cs cs)}) (Th taker)"
 (is "_ \<notin> ?R")
 proof -
-have "?R \<subseteq> ancestors (RAG s') (Th th')" by (rule ancestors_mono, auto)
+have "?R \<subseteq> ancestors (RAG s) (Th taker)" by (rule ancestors_mono, auto)
 moreover have "Th th1 \<notin> ..." using ancestors_th' assms by simp
 ultimately show ?thesis by auto
 qed
 next
-from assms show "Th th1 \<noteq> Th th'" by simp
+from assms show "Th th1 \<noteq> Th taker" by simp
 qed
 qed
 ultimately show ?thesis by (unfold RAG_s, simp)
 qed
 lemma cp_kept:
-assumes "th1 \<notin> {th, th'}"
+assumes "th1 \<notin> {th, taker}"
-shows "cp s th1 = cp s' th1"
+shows "cp (e#s) th1 = cp s th1"
-by (unfold cp_alt_def preced_kept subtree_kept[OF assms], simp)
+by (unfold cp_alt_def the_preced_es subtree_kept[OF assms], simp)
 end
-locale step_v_cps_nnt = step_v_cps +
-assumes nnt: "\<And> th'. (\<not> next_th s' th cs th')"
+context valid_trace_v_e
-context step_v_cps_nnt
 begin
-lemma RAG_s: "RAG s = RAG s' - {(Cs cs, Th th)}"
+find_theorems RAG s e
-proof -
-from nnt and  step_RAG_v[OF vt_s[unfolded s_def], folded s_def]
+lemma RAG_s: "RAG (e#s) = RAG s - {(Cs cs, Th th)}"
-show ?thesis by auto
+by (unfold RAG_es waiting_set_eq holding_set_eq, simp)
-qed
 lemma subtree_kept:
 assumes "th1 \<noteq> th"
-shows "subtree (RAG s) (Th th1) = subtree (RAG s') (Th th1)"
+shows "subtree (RAG (e#s)) (Th th1) = subtree (RAG s) (Th th1)"
 proof(unfold RAG_s, rule subset_del_subtree_outside)
-show "Range {(Cs cs, Th th)} \<inter> subtree (RAG s') (Th th1) = {}"
+show "Range {(Cs cs, Th th)} \<inter> subtree (RAG s) (Th th1) = {}"
 proof -
-have "(Th th) \<notin> subtree (RAG s') (Th th1)"
+have "(Th th) \<notin> subtree (RAG s) (Th th1)"
 proof(rule subtree_refute)
-show "Th th1 \<notin> ancestors (RAG s') (Th th)"
+show "Th th1 \<notin> ancestors (RAG s) (Th th)"
 by (unfold ancestors_th, simp)
 next
 from assms show "Th th1 \<noteq> Th th" by simp
 qed
 thus ?thesis by auto
 qed
 qed
 lemma cp_kept_1:
 assumes "th1 \<noteq> th"
-shows "cp s th1 = cp s' th1"
+shows "cp (e#s) th1 = cp s th1"
-by (unfold cp_alt_def preced_kept subtree_kept[OF assms], simp)
+by (unfold cp_alt_def the_preced_es subtree_kept[OF assms], simp)
-lemma subtree_cs: "subtree (RAG s') (Cs cs) = {Cs cs}"
+lemma subtree_cs: "subtree (RAG s) (Cs cs) = {Cs cs}"
 proof -
 { fix n
-have "(Cs cs) \<notin> ancestors (RAG s') n"
+have "(Cs cs) \<notin> ancestors (RAG s) n"
 proof
-assume "Cs cs \<in> ancestors (RAG s') n"
+assume "Cs cs \<in> ancestors (RAG s) n"
-hence "(n, Cs cs) \<in> (RAG s')^+" by (auto simp:ancestors_def)
+hence "(n, Cs cs) \<in> (RAG s)^+" by (auto simp:ancestors_def)
-from tranclE[OF this] obtain nn where h: "(nn, Cs cs) \<in> RAG s'" by auto
+from tranclE[OF this] obtain nn where h: "(nn, Cs cs) \<in> RAG s" by auto
 then obtain th' where "nn = Th th'"
 by (unfold s_RAG_def, auto)
-from h[unfolded this] have "(Th th', Cs cs) \<in> RAG s'" .
+from h[unfolded this] have "(Th th', Cs cs) \<in> RAG s" .
 from this[unfolded s_RAG_def]
-have "waiting (wq s') th' cs" by auto
+have "waiting (wq s) th' cs" by auto
 from this[unfolded cs_waiting_def]
-have "1 < length (wq s' cs)"
+have "1 < length (wq s cs)"
-by (cases "wq s' cs", auto)
+by (cases "wq s cs", auto)
-from holding_next_thI[OF holding_th this]
+from holding_next_thI[OF holding_th_cs_s this]
-obtain th' where "next_th s' th cs th'" by auto
+obtain th' where "next_th s th cs th'" by auto
-with nnt show False by auto
+thus False using no_taker by blast
 qed
 } note h = this
 {  fix n
-assume "n \<in> subtree (RAG s') (Cs cs)"
+assume "n \<in> subtree (RAG s) (Cs cs)"
 hence "n = (Cs cs)"
 by (elim subtreeE, insert h, auto)
-} moreover have "(Cs cs) \<in> subtree (RAG s') (Cs cs)"
+} moreover have "(Cs cs) \<in> subtree (RAG s) (Cs cs)"
 by (auto simp:subtree_def)
 ultimately show ?thesis by auto
 qed
 lemma subtree_th:
-"subtree (RAG s) (Th th) = subtree (RAG s') (Th th) - {Cs cs}"
+"subtree (RAG (e#s)) (Th th) = subtree (RAG s) (Th th) - {Cs cs}"
-proof(unfold RAG_s, fold subtree_cs, rule vat_s'.rtree_RAG.subtree_del_inside)
+proof(unfold RAG_s, fold subtree_cs, rule rtree_RAG.subtree_del_inside)
 from edge_of_th
-show "(Cs cs, Th th) \<in> edges_in (RAG s') (Th th)"
+show "(Cs cs, Th th) \<in> edges_in (RAG s) (Th th)"
 by (unfold edges_in_def, auto simp:subtree_def)
 qed
 lemma cp_kept_2:
-shows "cp s th = cp s' th"
+shows "cp (e#s) th = cp s th"
-by (unfold cp_alt_def subtree_th preced_kept, auto)
+by (unfold cp_alt_def subtree_th the_preced_es, auto)
 lemma eq_cp:
-shows "cp s th' = cp s' th'"
+shows "cp (e#s) th' = cp s th'"
 using cp_kept_1 cp_kept_2
 by (cases "th' = th", auto)
 end
-locale step_P_cps =
-fixes s' th cs s
-defines s_def : "s \<equiv> (P th cs#s')"
-assumes vt_s: "vt s"
-sublocale step_P_cps < vat_s : valid_trace "s"
-proof
-from vt_s show "vt s" .
-qed
 section {* The @{term P} operation *}
-sublocale step_P_cps < vat_s' : valid_trace "s'"
+context valid_trace_p
-proof
-from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
-qed
-context step_P_cps
 begin
-lemma readys_th: "th \<in> readys s'"
+lemma root_th: "root (RAG s) (Th th)"
-proof -
+by (simp add: ready_th_s readys_root)
-from step_back_step [OF vt_s[unfolded s_def]]
-have "PIP s' (P th cs)" .
-hence "th \<in> runing s'" by (cases, simp)
-thus ?thesis by (simp add:readys_def runing_def)
-qed
-lemma root_th: "root (RAG s') (Th th)"
-using readys_root[OF readys_th] .
 lemma in_no_others_subtree:
 assumes "th' \<noteq> th"
-shows "Th th \<notin> subtree (RAG s') (Th th')"
+shows "Th th \<notin> subtree (RAG s) (Th th')"
 proof
-assume "Th th \<in> subtree (RAG s') (Th th')"
+assume "Th th \<in> subtree (RAG s) (Th th')"
 thus False
 proof(cases rule:subtreeE)
 case 1
 with assms show ?thesis by auto
 next
 case 2
 with root_th show ?thesis by (auto simp:root_def)
 qed
 qed
-lemma preced_kept: "the_preced s = the_preced s'"
+lemma preced_kept: "the_preced (e#s) = the_preced s"
-by (auto simp: s_def the_preced_def preced_def)
+proof
+fix th'
+show "the_preced (e # s) th' = the_preced s th'"
+by (unfold the_preced_def is_p preced_def, simp)
+qed
 end
-locale step_P_cps_ne =step_P_cps +
-fixes th'
+context valid_trace_p_h
-assumes ne: "wq s' cs \<noteq> []"
-defines th'_def: "th' \<equiv> hd (wq s' cs)"
-locale step_P_cps_e =step_P_cps +
-assumes ee: "wq s' cs = []"
-context step_P_cps_e
 begin
-lemma 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 auto
-qed
 lemma subtree_kept:
 assumes "th' \<noteq> th"
-shows "subtree (RAG s) (Th th') = subtree (RAG s') (Th th')"
+shows "subtree (RAG (e#s)) (Th th') = subtree (RAG s) (Th th')"
-proof(unfold RAG_s, rule subtree_insert_next)
+proof(unfold RAG_es, rule subtree_insert_next)
 from in_no_others_subtree[OF assms]
-show "Th th \<notin> subtree (RAG s') (Th th')" .
+show "Th th \<notin> subtree (RAG s) (Th th')" .
 qed
 lemma cp_kept:
 assumes "th' \<noteq> th"
-shows "cp s th' = cp s' th'"
+shows "cp (e#s) th' = cp s th'"
 proof -
-have "(the_preced s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')}) =
+have "(the_preced (e#s) ` {th'a. Th th'a \<in> subtree (RAG (e#s)) (Th th')}) =
-(the_preced s' ` {th'a. Th th'a \<in> subtree (RAG s') (Th th')})"
+(the_preced s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')})"
 by (unfold preced_kept subtree_kept[OF assms], simp)
 thus ?thesis by (unfold cp_alt_def, simp)
 qed
 end
-context step_P_cps_ne
+context valid_trace_p_w
 begin
-lemma RAG_s: "RAG s = RAG s' \<union> {(Th th, Cs cs)}"
+interpretation vat_e: valid_trace "e#s"
-proof -
+by (unfold_locales, insert vt_e, simp)
-from step_RAG_p[OF vt_s[unfolded s_def]] and ne
-show ?thesis by (simp add:s_def)
+lemma cs_held: "(Cs cs, Th holder) \<in> RAG s"
-qed
+using holding_s_holder
+by (unfold s_RAG_def, fold holding_eq, auto)
-lemma cs_held: "(Cs cs, Th th') \<in> RAG s'"
-proof -
+lemma tRAG_s:
-have "(Cs cs, Th th') \<in> hRAG s'"
+"tRAG (e#s) = tRAG s \<union> {(Th th, Th holder)}"
+using local.RAG_tRAG_transfer[OF RAG_es cs_held] .
+lemma cp_kept:
+assumes "Th th'' \<notin> ancestors (tRAG (e#s)) (Th th)"
+shows "cp (e#s) th'' = cp s th''"
+proof -
+have h: "subtree (tRAG (e#s)) (Th th'') = subtree (tRAG s) (Th th'')"
 proof -
-from ne
+have "Th holder \<notin> subtree (tRAG s) (Th th'')"
-have " holding s' th' cs"
-by (unfold th'_def holding_eq cs_holding_def, auto)
-thus ?thesis
-by (unfold hRAG_def, auto)
-qed
-thus ?thesis by (unfold RAG_split, auto)
-qed
-lemma tRAG_s:
-"tRAG s = tRAG s' \<union> {(Th th, Th th')}"
-using RAG_tRAG_transfer[OF RAG_s cs_held] .
-lemma cp_kept:
-assumes "Th th'' \<notin> ancestors (tRAG s) (Th th)"
-shows "cp s th'' = cp s' th''"
-proof -
-have h: "subtree (tRAG s) (Th th'') = subtree (tRAG s') (Th th'')"
-proof -
-have "Th th' \<notin> subtree (tRAG s') (Th th'')"
 proof
-assume "Th th' \<in> subtree (tRAG s') (Th th'')"
+assume "Th holder \<in> subtree (tRAG s) (Th th'')"
 thus False
 proof(rule subtreeE)
-assume "Th th' = Th th''"
+assume "Th holder = Th th''"
 from assms[unfolded tRAG_s ancestors_def, folded this]
 show ?thesis by auto
 next
-assume "Th th'' \<in> ancestors (tRAG s') (Th th')"
+assume "Th th'' \<in> ancestors (tRAG s) (Th holder)"
-moreover have "... \<subseteq> ancestors (tRAG s) (Th th')"
+moreover have "... \<subseteq> ancestors (tRAG (e#s)) (Th holder)"
 proof(rule ancestors_mono)
-show "tRAG s' \<subseteq> tRAG s" by (unfold tRAG_s, auto)
+show "tRAG s \<subseteq> tRAG (e#s)" by (unfold tRAG_s, auto)
 qed
-ultimately have "Th th'' \<in> ancestors (tRAG s) (Th th')" by auto
+ultimately have "Th th'' \<in> ancestors (tRAG (e#s)) (Th holder)" by auto
-moreover have "Th th' \<in> ancestors (tRAG s) (Th th)"
+moreover have "Th holder \<in> ancestors (tRAG (e#s)) (Th th)"
 by (unfold tRAG_s, auto simp:ancestors_def)
-ultimately have "Th th'' \<in> ancestors (tRAG s) (Th th)"
+ultimately have "Th th'' \<in> ancestors (tRAG (e#s)) (Th th)"
 by (auto simp:ancestors_def)
 with assms show ?thesis by auto
 qed
 qed
 from subtree_insert_next[OF this]
-have "subtree (tRAG s' \<union> {(Th th, Th th')}) (Th th'') = subtree (tRAG s') (Th th'')" .
+have "subtree (tRAG s \<union> {(Th th, Th holder)}) (Th th'') = subtree (tRAG s) (Th th'')" .
 from this[folded tRAG_s] show ?thesis .
 qed
 show ?thesis by (unfold cp_alt_def1 h preced_kept, simp)
 qed
 lemma cp_gen_update_stop: (* ddd *)
-assumes "u \<in> ancestors (tRAG s) (Th th)"
+assumes "u \<in> ancestors (tRAG (e#s)) (Th th)"
-and "cp_gen s u = cp_gen s' u"
+and "cp_gen (e#s) u = cp_gen s u"
-and "y \<in> ancestors (tRAG s) u"
+and "y \<in> ancestors (tRAG (e#s)) u"
-shows "cp_gen s y = cp_gen s' y"
+shows "cp_gen (e#s) y = cp_gen s y"
 using assms(3)
-proof(induct rule:wf_induct[OF vat_s.fsbttRAGs.wf])
+proof(induct rule:wf_induct[OF vat_e.fsbttRAGs.wf])
 case (1 x)
 show ?case (is "?L = ?R")
 proof -
 from tRAG_ancestorsE[OF 1(2)]
 obtain th2 where eq_x: "x = Th th2" by blast
-from vat_s.cp_gen_rec[OF this]
+from vat_e.cp_gen_rec[OF this]
 have "?L =
-Max ({the_preced s th2} \<union> cp_gen s ` RTree.children (tRAG s) x)" .
+Max ({the_preced (e#s) th2} \<union> cp_gen (e#s) ` RTree.children (tRAG (e#s)) x)" .
 also have "... =
-Max ({the_preced s' th2} \<union> cp_gen s' ` RTree.children (tRAG s') x)"
+Max ({the_preced s th2} \<union> cp_gen s ` RTree.children (tRAG s) x)"
 proof -
-from preced_kept have "the_preced s th2 = the_preced s' th2" by simp
+from preced_kept have "the_preced (e#s) th2 = the_preced s th2" by simp
-moreover have "cp_gen s ` RTree.children (tRAG s) x =
+moreover have "cp_gen (e#s) ` RTree.children (tRAG (e#s)) x =
-cp_gen s' ` RTree.children (tRAG s') x"
+cp_gen s ` RTree.children (tRAG s) x"
 proof -
-have "RTree.children (tRAG s) x =  RTree.children (tRAG s') x"
+have "RTree.children (tRAG (e#s)) x =  RTree.children (tRAG s) x"
 proof(unfold tRAG_s, rule children_union_kept)
-have start: "(Th th, Th th') \<in> tRAG s"
+have start: "(Th th, Th holder) \<in> tRAG (e#s)"
 by (unfold tRAG_s, auto)
 note x_u = 1(2)
-show "x \<notin> Range {(Th th, Th th')}"
+show "x \<notin> Range {(Th th, Th holder)}"
 proof
-assume "x \<in> Range {(Th th, Th th')}"
+assume "x \<in> Range {(Th th, Th holder)}"
-hence eq_x: "x = Th th'" using RangeE by auto
+hence eq_x: "x = Th holder" using RangeE by auto
 show False
-proof(cases rule:vat_s.rtree_s.ancestors_headE[OF assms(1) start])
+proof(cases rule:vat_e.ancestors_headE[OF assms(1) start])
 case 1
-from x_u[folded this, unfolded eq_x] vat_s.acyclic_tRAG
+from x_u[folded this, unfolded eq_x] vat_e.acyclic_tRAG
 show ?thesis by (auto simp:ancestors_def acyclic_def)
 next
 case 2
 with x_u[unfolded eq_x]
-have "(Th th', Th th') \<in> (tRAG s)^+" by (auto simp:ancestors_def)
+have "(Th holder, Th holder) \<in> (tRAG (e#s))^+" by (auto simp:ancestors_def)
-with vat_s.acyclic_tRAG show ?thesis by (auto simp:acyclic_def)
+with vat_e.acyclic_tRAG show ?thesis by (auto simp:acyclic_def)
 qed
 qed
 qed
-moreover have "cp_gen s ` RTree.children (tRAG s) x =
+moreover have "cp_gen (e#s) ` RTree.children (tRAG (e#s)) x =
-cp_gen s' ` RTree.children (tRAG s) x" (is "?f ` ?A = ?g ` ?A")
+cp_gen s ` RTree.children (tRAG (e#s)) x" (is "?f ` ?A = ?g ` ?A")
 proof(rule f_image_eq)
 fix a
 assume a_in: "a \<in> ?A"
 from 1(2)
 show "?f a = ?g a"
-proof(cases rule:vat_s.rtree_s.ancestors_childrenE[case_names in_ch out_ch])
+proof(cases rule:vat_e.rtree_s.ancestors_childrenE[case_names in_ch out_ch])
 case in_ch
 show ?thesis
 proof(cases "a = u")
 case True
 from assms(2)[folded this] show ?thesis .
 next
 case False
-have a_not_in: "a \<notin> ancestors (tRAG s) (Th th)"
+have a_not_in: "a \<notin> ancestors (tRAG (e#s)) (Th th)"
 proof
-assume a_in': "a \<in> ancestors (tRAG s) (Th th)"
+assume a_in': "a \<in> ancestors (tRAG (e#s)) (Th th)"
 have "a = u"
-proof(rule vat_s.rtree_s.ancestors_children_unique)
+proof(rule vat_e.rtree_s.ancestors_children_unique)
-from a_in' a_in show "a \<in> ancestors (tRAG s) (Th th) \<inter>
+from a_in' a_in show "a \<in> ancestors (tRAG (e#s)) (Th th) \<inter>
-RTree.children (tRAG s) x" by auto
+RTree.children (tRAG (e#s)) x" by auto
 next
-from assms(1) in_ch show "u \<in> ancestors (tRAG s) (Th th) \<inter>
+from assms(1) in_ch show "u \<in> ancestors (tRAG (e#s)) (Th th) \<inter>
-RTree.children (tRAG s) x" by auto
+RTree.children (tRAG (e#s)) x" by auto
 qed
 with False show False by simp
 qed
 from a_in obtain th_a where eq_a: "a = Th th_a"
 by (unfold RTree.children_def tRAG_alt_def, auto)
 from cp_kept[OF a_not_in[unfolded eq_a]]
-have "cp s th_a = cp s' th_a" .
+have "cp (e#s) th_a = cp s th_a" .
 from this [unfolded cp_gen_def_cond[OF eq_a], folded eq_a]
 show ?thesis .
 qed
 next
 case (out_ch z)
-hence h: "z \<in> ancestors (tRAG s) u" "z \<in> RTree.children (tRAG s) x" by auto
+hence h: "z \<in> ancestors (tRAG (e#s)) u" "z \<in> RTree.children (tRAG (e#s)) x" by auto
 show ?thesis
 proof(cases "a = z")
 case True
-from h(2) have zx_in: "(z, x) \<in> (tRAG s)" by (auto simp:RTree.children_def)
+from h(2) have zx_in: "(z, x) \<in> (tRAG (e#s))" by (auto simp:RTree.children_def)
 from 1(1)[rule_format, OF this h(1)]
-have eq_cp_gen: "cp_gen s z = cp_gen s' z" .
+have eq_cp_gen: "cp_gen (e#s) z = cp_gen s z" .
 with True show ?thesis by metis
 next
 case False
 from a_in obtain th_a where eq_a: "a = Th th_a"
 by (auto simp:RTree.children_def tRAG_alt_def)
-have "a \<notin> ancestors (tRAG s) (Th th)"
+have "a \<notin> ancestors (tRAG (e#s)) (Th th)"
 proof
-assume a_in': "a \<in> ancestors (tRAG s) (Th th)"
+assume a_in': "a \<in> ancestors (tRAG (e#s)) (Th th)"
 have "a = z"
-proof(rule vat_s.rtree_s.ancestors_children_unique)
+proof(rule vat_e.rtree_s.ancestors_children_unique)
-from assms(1) h(1) have "z \<in> ancestors (tRAG s) (Th th)"
+from assms(1) h(1) have "z \<in> ancestors (tRAG (e#s)) (Th th)"
 by (auto simp:ancestors_def)
-with h(2) show " z \<in> ancestors (tRAG s) (Th th) \<inter>
+with h(2) show " z \<in> ancestors (tRAG (e#s)) (Th th) \<inter>
-RTree.children (tRAG s) x" by auto
+RTree.children (tRAG (e#s)) x" by auto
 next
 from a_in a_in'
-show "a \<in> ancestors (tRAG s) (Th th) \<inter> RTree.children (tRAG s) x"
+show "a \<in> ancestors (tRAG (e#s)) (Th th) \<inter> RTree.children (tRAG (e#s)) x"
 by auto
 qed
 with False show False by auto
 qed
 from cp_kept[OF this[unfolded eq_a]]
-have "cp s th_a = cp s' th_a" .
+have "cp (e#s) th_a = cp s th_a" .
 from this[unfolded cp_gen_def_cond[OF eq_a], folded eq_a]
 show ?thesis .
 qed
 qed
 qed
 ultimately show ?thesis by metis
 qed
 ultimately show ?thesis by simp
 qed
 also have "... = ?R"
-by (fold vat_s'.cp_gen_rec[OF eq_x], simp)
+by (fold cp_gen_rec[OF eq_x], simp)
 finally show ?thesis .
 qed
 qed
 lemma cp_up:
-assumes "(Th th') \<in> ancestors (tRAG s) (Th th)"
+assumes "(Th th') \<in> ancestors (tRAG (e#s)) (Th th)"
-and "cp s th' = cp s' th'"
+and "cp (e#s) th' = cp s th'"
-and "(Th th'') \<in> ancestors (tRAG s) (Th th')"
+and "(Th th'') \<in> ancestors (tRAG (e#s)) (Th th')"
-shows "cp s th'' = cp s' th''"
+shows "cp (e#s) th'' = cp s th''"
 proof -
-have "cp_gen s (Th th'') = cp_gen s' (Th th'')"
+have "cp_gen (e#s) (Th th'') = cp_gen s (Th th'')"
 proof(rule cp_gen_update_stop[OF assms(1) _ assms(3)])
 from assms(2) cp_gen_def_cond[OF refl[of "Th th'"]]
-show "cp_gen s (Th th') = cp_gen s' (Th th')" by metis
+show "cp_gen (e#s) (Th th') = cp_gen s (Th th')" by metis
 qed
 with cp_gen_def_cond[OF refl[of "Th th''"]]
 show ?thesis by metis
 qed
 end
 section {* The @{term Create} operation *}
-locale step_create_cps =
+context valid_trace_create
-fixes s' th prio s
+begin
-defines s_def : "s \<equiv> (Create th prio#s')"
-assumes vt_s: "vt s"
+interpretation vat_e: valid_trace "e#s"
+by (unfold_locales, insert vt_e, simp)
-sublocale step_create_cps < vat_s: valid_trace "s"
-by (unfold_locales, insert vt_s, simp)
+lemma tRAG_kept: "tRAG (e#s) = tRAG s"
+by (unfold tRAG_alt_def RAG_unchanged, auto)
-sublocale step_create_cps < vat_s': valid_trace "s'"
-by (unfold_locales, insert step_back_vt[OF vt_s[unfolded s_def]], simp)
-context step_create_cps
-begin
-lemma RAG_kept: "RAG s = RAG s'"
-by (unfold s_def RAG_create_unchanged, auto)
-lemma tRAG_kept: "tRAG s = tRAG s'"
-by (unfold tRAG_alt_def RAG_kept, auto)
 lemma preced_kept:
 assumes "th' \<noteq> th"
-shows "the_preced s th' = the_preced s' th'"
+shows "the_preced (e#s) th' = the_preced s th'"
-by (unfold s_def the_preced_def preced_def, insert assms, auto)
+by (unfold the_preced_def preced_def is_create, insert assms, auto)
-lemma th_not_in: "Th th \<notin> Field (tRAG s')"
+lemma th_not_in: "Th th \<notin> Field (tRAG s)"
-proof -
+by (meson not_in_thread_isolated subsetCE tRAG_Field th_not_live_s)
-from vt_s[unfolded s_def]
-have "PIP s' (Create th prio)" by (cases, simp)
-hence "th \<notin> threads s'" by(cases, simp)
-from vat_s'.not_in_thread_isolated[OF this]
-have "Th th \<notin> Field (RAG s')" .
-with tRAG_Field show ?thesis by auto
-qed
 lemma eq_cp:
 assumes neq_th: "th' \<noteq> th"
-shows "cp s th' = cp s' th'"
+shows "cp (e#s) th' = cp s th'"
 proof -
-have "(the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th') =
+have "(the_preced (e#s) \<circ> the_thread) ` subtree (tRAG (e#s)) (Th th') =
-(the_preced s' \<circ> the_thread) ` subtree (tRAG s') (Th th')"
+(the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th')"
 proof(unfold tRAG_kept, rule f_image_eq)
 fix a
-assume a_in: "a \<in> subtree (tRAG s') (Th th')"
+assume a_in: "a \<in> subtree (tRAG s) (Th th')"
 then obtain th_a where eq_a: "a = Th th_a"
 proof(cases rule:subtreeE)
 case 2
 from ancestors_Field[OF 2(2)]
 and that show ?thesis by (unfold tRAG_alt_def, auto)
 qed auto
 have neq_th_a: "th_a \<noteq> th"
 proof -
-have "(Th th) \<notin> subtree (tRAG s') (Th th')"
+have "(Th th) \<notin> subtree (tRAG s) (Th th')"
 proof
-assume "Th th \<in> subtree (tRAG s') (Th th')"
+assume "Th th \<in> subtree (tRAG s) (Th th')"
 thus False
 proof(cases rule:subtreeE)
 case 2
 from ancestors_Field[OF this(2)]
 and th_not_in[unfolded Field_def]
 qed (insert assms, auto)
 qed
 with a_in[unfolded eq_a] show ?thesis by auto
 qed
 from preced_kept[OF this]
-show "(the_preced s \<circ> the_thread) a = (the_preced s' \<circ> the_thread) a"
+show "(the_preced (e#s) \<circ> the_thread) a = (the_preced s \<circ> the_thread) a"
 by (unfold eq_a, simp)
 qed
 thus ?thesis by (unfold cp_alt_def1, simp)
 qed
-lemma children_of_th: "RTree.children (tRAG s) (Th th) = {}"
+lemma children_of_th: "RTree.children (tRAG (e#s)) (Th th) = {}"
 proof -
 { fix a
-assume "a \<in> RTree.children (tRAG s) (Th th)"
+assume "a \<in> RTree.children (tRAG (e#s)) (Th th)"
-hence "(a, Th th) \<in> tRAG s" by (auto simp:RTree.children_def)
+hence "(a, Th th) \<in> tRAG (e#s)" by (auto simp:RTree.children_def)
 with th_not_in have False
 by (unfold Field_def tRAG_kept, auto)
 } thus ?thesis by auto
 qed
-lemma eq_cp_th: "cp s th = preced th s"
+lemma eq_cp_th: "cp (e#s) th = preced th (e#s)"
-by (unfold vat_s.cp_rec children_of_th, simp add:the_preced_def)
+by (unfold vat_e.cp_rec children_of_th, simp add:the_preced_def)
 end
-locale step_exit_cps =
-fixes s' th prio s
+context valid_trace_exit
-defines s_def : "s \<equiv> Exit th # s'"
-assumes vt_s: "vt s"
-sublocale step_exit_cps < vat_s: valid_trace "s"
-by (unfold_locales, insert vt_s, simp)
-sublocale step_exit_cps < vat_s': valid_trace "s'"
-by (unfold_locales, insert step_back_vt[OF vt_s[unfolded s_def]], simp)
-context step_exit_cps
 begin
 lemma preced_kept:
 assumes "th' \<noteq> th"
-shows "the_preced s th' = the_preced s' th'"
+shows "the_preced (e#s) th' = the_preced s th'"
-by (unfold s_def the_preced_def preced_def, insert assms, auto)
+using assms
+by (unfold the_preced_def is_exit preced_def, simp)
-lemma RAG_kept: "RAG s = RAG s'"
-by (unfold s_def RAG_exit_unchanged, auto)
+lemma tRAG_kept: "tRAG (e#s) = tRAG s"
+by (unfold tRAG_alt_def RAG_unchanged, auto)
-lemma tRAG_kept: "tRAG s = tRAG s'"
-by (unfold tRAG_alt_def RAG_kept, auto)
+lemma th_RAG: "Th th \<notin> Field (RAG s)"
+proof -
-lemma th_ready: "th \<in> readys s'"
+have "Th th \<notin> Range (RAG s)"
-proof -
-from vt_s[unfolded s_def]
-have "PIP s' (Exit th)" by (cases, simp)
-hence h: "th \<in> runing s' \<and> holdents s' th = {}" by (cases, metis)
-thus ?thesis by (unfold runing_def, auto)
-qed
-lemma th_holdents: "holdents s' th = {}"
-proof -
-from vt_s[unfolded s_def]
-have "PIP s' (Exit th)" by (cases, simp)
-thus ?thesis by (cases, metis)
-qed
-lemma th_RAG: "Th th \<notin> Field (RAG s')"
-proof -
-have "Th th \<notin> Range (RAG s')"
 proof
-assume "Th th \<in> Range (RAG s')"
+assume "Th th \<in> Range (RAG s)"
-then obtain cs where "holding (wq s') th cs"
+then obtain cs where "holding (wq s) th cs"
 by (unfold Range_iff s_RAG_def, auto)
-with th_holdents[unfolded holdents_def]
+with holdents_th_s[unfolded holdents_def]
-show False by (unfold eq_holding, auto)
+show False by (unfold holding_eq, auto)
 qed
-moreover have "Th th \<notin> Domain (RAG s')"
+moreover have "Th th \<notin> Domain (RAG s)"
 proof
-assume "Th th \<in> Domain (RAG s')"
+assume "Th th \<in> Domain (RAG s)"
-then obtain cs where "waiting (wq s') th cs"
+then obtain cs where "waiting (wq s) th cs"
 by (unfold Domain_iff s_RAG_def, auto)
-with th_ready show False by (unfold readys_def eq_waiting, auto)
+with th_ready_s show False by (unfold readys_def waiting_eq, auto)
 qed
 ultimately show ?thesis by (auto simp:Field_def)
 qed
-lemma th_tRAG: "(Th th) \<notin> Field (tRAG s')"
+lemma th_tRAG: "(Th th) \<notin> Field (tRAG s)"
-using th_RAG tRAG_Field[of s'] by auto
+using th_RAG tRAG_Field by auto
 lemma eq_cp:
 assumes neq_th: "th' \<noteq> th"
-shows "cp s th' = cp s' th'"
+shows "cp (e#s) th' = cp s th'"
 proof -
-have "(the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th') =
+have "(the_preced (e#s) \<circ> the_thread) ` subtree (tRAG (e#s)) (Th th') =
-(the_preced s' \<circ> the_thread) ` subtree (tRAG s') (Th th')"
+(the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th')"
 proof(unfold tRAG_kept, rule f_image_eq)
 fix a
-assume a_in: "a \<in> subtree (tRAG s') (Th th')"
+assume a_in: "a \<in> subtree (tRAG s) (Th th')"
 then obtain th_a where eq_a: "a = Th th_a"
 proof(cases rule:subtreeE)
 case 2
 from ancestors_Field[OF 2(2)]
 and that show ?thesis by (unfold tRAG_alt_def, auto)
 qed auto
 have neq_th_a: "th_a \<noteq> th"
 proof -
-from vat_s'.readys_in_no_subtree[OF th_ready assms]
+from readys_in_no_subtree[OF th_ready_s assms]
-have "(Th th) \<notin> subtree (RAG s') (Th th')" .
+have "(Th th) \<notin> subtree (RAG s) (Th th')" .
-with tRAG_subtree_RAG[of s' "Th th'"]
+with tRAG_subtree_RAG[of s "Th th'"]
-have "(Th th) \<notin> subtree (tRAG s') (Th th')" by auto
+have "(Th th) \<notin> subtree (tRAG s) (Th th')" by auto
 with a_in[unfolded eq_a] show ?thesis by auto
 qed
 from preced_kept[OF this]
-show "(the_preced s \<circ> the_thread) a = (the_preced s' \<circ> the_thread) a"
+show "(the_preced (e#s) \<circ> the_thread) a = (the_preced s \<circ> the_thread) a"
 by (unfold eq_a, simp)
 qed
 thus ?thesis by (unfold cp_alt_def1, simp)
 qed
 end
 end
+=======
+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

changeset 92	4763aa246dbd
parent 68	db196b066b97
child 93	524bd3caa6b6