--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/ExtGG.thy Thu Jan 28 07:43:05 2016 +0800
@@ -0,0 +1,708 @@
+section {*
+ This file contains lemmas used to guide the recalculation of current precedence
+ after every system call (or system operation)
+*}
+theory ExtGG
+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
+ 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 every threads in its subtree,
+ where the notion of sub-tree in RAG is defined in RTree.thy.
+
+ Therefore, for each operation, lemmas about the change of precedences
+ and RAG are derived first, on which lemmas about current precedence
+ recalculation are based on.
+*}
+
+section {* The @{term Set} operation *}
+
+context valid_trace_set
+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' (e#s) = preced th' s"
+proof -
+ from assms show ?thesis
+ by (unfold is_set, auto simp:preced_def)
+qed
+
+lemma eq_the_preced:
+ assumes "th' \<noteq> th"
+ shows "the_preced (e#s) th' = the_preced s th'"
+ using assms
+ by (unfold the_preced_def, intro eq_preced, simp)
+
+
+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')"
+ 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 (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')})"
+ (is "Max (?f ` ?S1) = Max (?g ` ?S2)")
+ proof -
+ -- {* The base sets are equal. *}
+ 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 (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
+ thus ?thesis by (simp add:cp_alt_def)
+qed
+
+text {*
+ 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')"
+proof -
+ from readys_in_no_subtree[OF th_ready_s 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 (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.
+*}
+
+
+context valid_trace_v
+begin
+
+lemma ancestors_th: "ancestors (RAG s) (Th th) = {}"
+proof -
+ from readys_root[OF th_ready_s]
+ show ?thesis
+ by (unfold root_def, simp)
+qed
+
+lemma edge_of_th:
+ "(Cs cs, Th th) \<in> RAG s"
+proof -
+ 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}"
+proof -
+ have "ancestors (RAG s) (Cs cs) = ancestors (RAG s) (Th th) \<union> {Th th}"
+ by (rule rtree_RAG.ancestors_accum[OF edge_of_th])
+ from this[unfolded ancestors_th] show ?thesis by simp
+qed
+
+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"}.
+*}
+
+context valid_trace_v_n
+begin
+
+lemma sub_RAGs':
+ "{(Cs cs, Th th), (Th taker, Cs cs)} \<subseteq> RAG s"
+ using next_th_RAG[OF next_th_taker] .
+
+lemma ancestors_th':
+ "ancestors (RAG s) (Th taker) = {Th th, Cs cs}"
+proof -
+ have "ancestors (RAG s) (Th taker) = ancestors (RAG s) (Cs cs) \<union> {Cs cs}"
+ proof(rule rtree_RAG.ancestors_accum)
+ 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 (e#s) = (RAG s - {(Cs cs, Th th), (Th taker, Cs cs)}) \<union>
+ {(Cs cs, Th taker)}"
+ by (unfold RAG_es waiting_set_eq holding_set_eq, auto)
+
+lemma subtree_kept: (* ddd *)
+ assumes "th1 \<notin> {th, taker}"
+ shows "subtree (RAG (e#s)) (Th th1) =
+ subtree (RAG s) (Th th1)" (is "_ = ?R")
+proof -
+ 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 taker, Cs cs)} \<inter> subtree (RAG s) (Th th1) = {}"
+ proof -
+ have "(Th th) \<notin> subtree (RAG s) (Th th1)"
+ proof(rule subtree_refute)
+ 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)"
+ proof(rule subtree_refute)
+ 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
+ thus ?thesis by simp
+ qed
+ qed
+ moreover have "subtree ?RAG'' (Th th1) = subtree ?RAG' (Th th1)"
+ proof(rule subtree_insert_next)
+ 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 taker, Cs cs)}) (Th taker)"
+ (is "_ \<notin> ?R")
+ proof -
+ 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 taker" by simp
+ qed
+ qed
+ ultimately show ?thesis by (unfold RAG_s, simp)
+qed
+
+lemma cp_kept:
+ assumes "th1 \<notin> {th, taker}"
+ shows "cp (e#s) th1 = cp s th1"
+ by (unfold cp_alt_def the_preced_es subtree_kept[OF assms], simp)
+
+end
+
+
+context valid_trace_v_e
+begin
+
+find_theorems RAG s e
+
+lemma RAG_s: "RAG (e#s) = RAG s - {(Cs cs, Th th)}"
+ by (unfold RAG_es waiting_set_eq holding_set_eq, simp)
+
+lemma subtree_kept:
+ assumes "th1 \<noteq> th"
+ 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) = {}"
+ proof -
+ have "(Th th) \<notin> subtree (RAG s) (Th th1)"
+ proof(rule subtree_refute)
+ 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 (e#s) th1 = cp s th1"
+ by (unfold cp_alt_def the_preced_es subtree_kept[OF assms], simp)
+
+lemma subtree_cs: "subtree (RAG s) (Cs cs) = {Cs cs}"
+proof -
+ { fix n
+ have "(Cs cs) \<notin> ancestors (RAG s) n"
+ proof
+ assume "Cs cs \<in> ancestors (RAG s) n"
+ 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
+ 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 this[unfolded s_RAG_def]
+ have "waiting (wq s) th' cs" by auto
+ from this[unfolded cs_waiting_def]
+ have "1 < length (wq s cs)"
+ by (cases "wq s cs", auto)
+ from holding_next_thI[OF holding_th_cs_s this]
+ obtain th' where "next_th s th cs th'" by auto
+ thus False using no_taker by blast
+ qed
+ } note h = this
+ { fix n
+ 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)"
+ by (auto simp:subtree_def)
+ ultimately show ?thesis by auto
+qed
+
+lemma subtree_th:
+ "subtree (RAG (e#s)) (Th th) = subtree (RAG s) (Th th) - {Cs cs}"
+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)"
+ by (unfold edges_in_def, auto simp:subtree_def)
+qed
+
+lemma cp_kept_2:
+ shows "cp (e#s) th = cp s th"
+ by (unfold cp_alt_def subtree_th the_preced_es, auto)
+
+lemma eq_cp:
+ shows "cp (e#s) th' = cp s th'"
+ using cp_kept_1 cp_kept_2
+ by (cases "th' = th", auto)
+
+end
+
+
+section {* The @{term P} operation *}
+
+context valid_trace_p
+begin
+
+lemma root_th: "root (RAG s) (Th th)"
+ by (simp add: ready_th_s readys_root)
+
+lemma in_no_others_subtree:
+ assumes "th' \<noteq> th"
+ shows "Th th \<notin> subtree (RAG s) (Th th')"
+proof
+ assume "Th th \<in> subtree (RAG s) (Th th')"
+ thus False
+ proof(cases rule:subtreeE)
+ case 1
+ with assms show ?thesis by auto
+ next
+ case 2
+ with root_th show ?thesis by (auto simp:root_def)
+ qed
+qed
+
+lemma preced_kept: "the_preced (e#s) = the_preced s"
+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
+
+
+context valid_trace_p_h
+begin
+
+lemma subtree_kept:
+ assumes "th' \<noteq> th"
+ shows "subtree (RAG (e#s)) (Th th') = subtree (RAG s) (Th th')"
+proof(unfold RAG_es, rule subtree_insert_next)
+ from in_no_others_subtree[OF assms]
+ show "Th th \<notin> subtree (RAG s) (Th th')" .
+qed
+
+lemma cp_kept:
+ assumes "th' \<noteq> th"
+ shows "cp (e#s) th' = cp s th'"
+proof -
+ 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')})"
+ by (unfold preced_kept subtree_kept[OF assms], simp)
+ thus ?thesis by (unfold cp_alt_def, simp)
+qed
+
+end
+
+context valid_trace_p_w
+begin
+
+interpretation vat_e: valid_trace "e#s"
+ by (unfold_locales, insert vt_e, simp)
+
+lemma cs_held: "(Cs cs, Th holder) \<in> RAG s"
+ using holding_s_holder
+ by (unfold s_RAG_def, fold holding_eq, auto)
+
+lemma tRAG_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 -
+ have "Th holder \<notin> subtree (tRAG s) (Th th'')"
+ proof
+ assume "Th holder \<in> subtree (tRAG s) (Th th'')"
+ thus False
+ proof(rule subtreeE)
+ 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 holder)"
+ moreover have "... \<subseteq> ancestors (tRAG (e#s)) (Th holder)"
+ proof(rule ancestors_mono)
+ show "tRAG s \<subseteq> tRAG (e#s)" by (unfold tRAG_s, auto)
+ qed
+ ultimately have "Th th'' \<in> ancestors (tRAG (e#s)) (Th holder)" by auto
+ 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 (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 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 (e#s)) (Th th)"
+ and "cp_gen (e#s) u = cp_gen s u"
+ and "y \<in> ancestors (tRAG (e#s)) u"
+ shows "cp_gen (e#s) y = cp_gen s y"
+ using assms(3)
+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_e.cp_gen_rec[OF this]
+ have "?L =
+ 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)"
+ proof -
+ from preced_kept have "the_preced (e#s) th2 = the_preced s th2" by simp
+ moreover have "cp_gen (e#s) ` RTree.children (tRAG (e#s)) x =
+ cp_gen s ` RTree.children (tRAG s) x"
+ proof -
+ 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 holder) \<in> tRAG (e#s)"
+ by (unfold tRAG_s, auto)
+ note x_u = 1(2)
+ show "x \<notin> Range {(Th th, Th holder)}"
+ proof
+ assume "x \<in> Range {(Th th, Th holder)}"
+ hence eq_x: "x = Th holder" using RangeE by auto
+ show False
+ proof(cases rule:vat_e.ancestors_headE[OF assms(1) start])
+ case 1
+ 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 holder, Th holder) \<in> (tRAG (e#s))^+" by (auto simp:ancestors_def)
+ with vat_e.acyclic_tRAG show ?thesis by (auto simp:acyclic_def)
+ qed
+ qed
+ qed
+ moreover have "cp_gen (e#s) ` RTree.children (tRAG (e#s)) x =
+ 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_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 (e#s)) (Th th)"
+ proof
+ assume a_in': "a \<in> ancestors (tRAG (e#s)) (Th th)"
+ have "a = u"
+ proof(rule vat_e.rtree_s.ancestors_children_unique)
+ from a_in' a_in show "a \<in> ancestors (tRAG (e#s)) (Th th) \<inter>
+ RTree.children (tRAG (e#s)) x" by auto
+ next
+ from assms(1) in_ch show "u \<in> ancestors (tRAG (e#s)) (Th th) \<inter>
+ 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 (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 (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 (e#s))" by (auto simp:RTree.children_def)
+ from 1(1)[rule_format, OF this h(1)]
+ 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 (e#s)) (Th th)"
+ proof
+ assume a_in': "a \<in> ancestors (tRAG (e#s)) (Th th)"
+ have "a = z"
+ proof(rule vat_e.rtree_s.ancestors_children_unique)
+ 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 (e#s)) (Th th) \<inter>
+ RTree.children (tRAG (e#s)) x" by auto
+ next
+ from a_in a_in'
+ 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 (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 cp_gen_rec[OF eq_x], simp)
+ finally show ?thesis .
+ qed
+qed
+
+lemma cp_up:
+ assumes "(Th th') \<in> ancestors (tRAG (e#s)) (Th th)"
+ and "cp (e#s) th' = cp s th'"
+ and "(Th th'') \<in> ancestors (tRAG (e#s)) (Th th')"
+ shows "cp (e#s) th'' = cp s th''"
+proof -
+ 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 (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 *}
+
+context valid_trace_create
+begin
+
+interpretation vat_e: valid_trace "e#s"
+ by (unfold_locales, insert vt_e, simp)
+
+lemma tRAG_kept: "tRAG (e#s) = tRAG s"
+ by (unfold tRAG_alt_def RAG_unchanged, auto)
+
+lemma preced_kept:
+ assumes "th' \<noteq> th"
+ shows "the_preced (e#s) th' = the_preced s th'"
+ by (unfold the_preced_def preced_def is_create, insert assms, auto)
+
+lemma th_not_in: "Th th \<notin> Field (tRAG s)"
+ by (meson not_in_thread_isolated subsetCE tRAG_Field th_not_live_s)
+
+lemma eq_cp:
+ assumes neq_th: "th' \<noteq> th"
+ shows "cp (e#s) th' = cp s th'"
+proof -
+ 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')"
+ proof(unfold tRAG_kept, rule f_image_eq)
+ fix a
+ 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')"
+ proof
+ 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]
+ show ?thesis by auto
+ qed (insert assms, auto)
+ qed
+ with a_in[unfolded eq_a] show ?thesis by auto
+ qed
+ from preced_kept[OF this]
+ 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 (e#s)) (Th th) = {}"
+proof -
+ { fix a
+ assume "a \<in> RTree.children (tRAG (e#s)) (Th th)"
+ 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 (e#s) th = preced th (e#s)"
+ by (unfold vat_e.cp_rec children_of_th, simp add:the_preced_def)
+
+end
+
+
+context valid_trace_exit
+begin
+
+lemma preced_kept:
+ assumes "th' \<noteq> th"
+ shows "the_preced (e#s) th' = the_preced s th'"
+ using assms
+ by (unfold the_preced_def is_exit preced_def, simp)
+
+lemma tRAG_kept: "tRAG (e#s) = tRAG s"
+ by (unfold tRAG_alt_def RAG_unchanged, auto)
+
+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)"
+ then obtain cs where "holding (wq s) th cs"
+ by (unfold Range_iff s_RAG_def, auto)
+ with holdents_th_s[unfolded holdents_def]
+ show False by (unfold holding_eq, auto)
+ qed
+ moreover have "Th th \<notin> Domain (RAG s)"
+ proof
+ assume "Th th \<in> Domain (RAG s)"
+ then obtain cs where "waiting (wq s) th cs"
+ by (unfold Domain_iff s_RAG_def, 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)"
+ using th_RAG tRAG_Field by auto
+
+lemma eq_cp:
+ assumes neq_th: "th' \<noteq> th"
+ shows "cp (e#s) th' = cp s th'"
+proof -
+ 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')"
+ proof(unfold tRAG_kept, rule f_image_eq)
+ fix a
+ 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 readys_in_no_subtree[OF th_ready_s assms]
+ have "(Th th) \<notin> subtree (RAG s) (Th th')" .
+ with tRAG_subtree_RAG[of s "Th th'"]
+ have "(Th th) \<notin> subtree (tRAG s) (Th th')" by auto
+ with a_in[unfolded eq_a] show ?thesis by auto
+ qed
+ from preced_kept[OF this]
+ 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
+