--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/ExtGG.thy Fri Jun 17 09:46:25 2016 +0100
@@ -0,0 +1,702 @@
+section {*
+ This file contains lemmas used to guide the recalculation of current precedence
+ after every system call (or system operation)
+*}
+theory Implementation
+imports PIPBasics
+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
+
+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_es.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_es.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_es.ancestors_headE[OF assms(1) start])
+ case 1
+ from x_u[folded this, unfolded eq_x] vat_es.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_es.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_es.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_es.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_es.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
+
+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_es.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
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/PrioG.thy Fri Jun 17 09:46:25 2016 +0100
@@ -0,0 +1,797 @@
+theory Correctness
+imports PIPBasics
+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
+
+lemma highest_cp_preced: "cp s th = Max (the_preced s ` threads s)"
+ using eq_cp_s_th highest max_cp_eq the_preced_def by presburger
+
+
+lemma highest_preced_thread: "preced th s = Max (the_preced s ` threads s)"
+ by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma highest': "cp s th = Max (cp s ` threads s)"
+ by (simp add: eq_cp_s_th highest)
+
+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 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[simp]: "Max (cp (t@s) ` threads (t@s)) = cp (t@s) th"
+ using max_cp_eq th_cp_max_preced the_preced_def vt_t by presburger
+
+lemma [simp]: "Max (cp (t@s) ` threads (t@s)) = Max (the_preced (t@s) ` threads (t@s))"
+ by (simp add: th_cp_max_preced)
+
+lemma [simp]: "Max (the_preced (t@s) ` threads (t@s)) = the_preced (t@s) th"
+ using max_kept th_kept the_preced_def by auto
+
+lemma [simp]: "the_preced (t@s) th = preced th (t@s)"
+ using the_preced_def by auto
+
+lemma [simp]: "preced th (t@s) = preced th s"
+ by (simp add: th_kept)
+
+lemma [simp]: "cp s th = preced th s"
+ by (simp add: eq_cp_s_th)
+
+lemma th_cp_preced [simp]: "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)
+
+section {* The `blocking thread` *}
+
+text {*
+ The purpose of PIP is to ensure that the most
+ urgent thread @{term th} is not blocked unreasonably.
+ Therefore, a clear picture of the blocking thread is essential
+ to assure people that the purpose is fulfilled.
+
+ In this section, we are going to derive a series of lemmas
+ with finally give rise to a picture of the blocking thread.
+
+ By `blocking thread`, we mean a thread in running state but
+ different from thread @{term th}.
+*}
+
+text {*
+ The following lemmas shows that the @{term cp}-value
+ of the blocking thread @{text th'} equals to the highest
+ precedence in the whole system.
+*}
+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 following lemma shows how the counters for @{term "P"} and
+ @{term "V"} operations relate to the running threads in the states
+ @{term s} and @{term "t @ s"}. The lemma shows that if a thread's
+ @{term "P"}-count equals its @{term "V"}-count (which means it no
+ longer has any resource in its possession), it cannot be a running
+ thread.
+
+ The proof is by contraction with the assumption @{text "th' \<noteq> th"}.
+ The key is the use of @{thm eq_pv_dependants} to derive the
+ emptiness of @{text th'}s @{term dependants}-set from the balance of
+ its @{term P} and @{term V} counts. From this, it can be shown
+ @{text th'}s @{term cp}-value equals to its own precedence.
+
+ On the other hand, since @{text th'} is running, by @{thm
+ runing_preced_inversion}, its @{term cp}-value equals to the
+ precedence of @{term th}.
+
+ Combining the above two resukts we have that @{text th'} and @{term
+ th} have the same precedence. By uniqueness of precedences, we have
+ @{text "th' = th"}, which is in contradiction with the assumption
+ @{text "th' \<noteq> th"}.
+
+*}
+
+lemma eq_pv_blocked: (* ddd *)
+ assumes 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'_in: "th' \<in> threads (t@s)"
+ using otherwise readys_threads runing_def by auto
+ have "th' = th"
+ proof(rule preced_unique)
+ -- {* The proof goes like this:
+ it is first shown that the @{term preced}-value of @{term th'}
+ equals to that of @{term th}, then by uniqueness
+ of @{term preced}-values (given by lemma @{thm preced_unique}),
+ @{term th'} equals to @{term th}: *}
+ show "preced th' (t @ s) = preced th (t @ s)" (is "?L = ?R")
+ proof -
+ -- {* Since the counts of @{term th'} are balanced, the subtree
+ of it contains only itself, so, its @{term cp}-value
+ equals its @{term preced}-value: *}
+ have "?L = cp (t@s) th'"
+ by (unfold cp_eq_cpreced cpreced_def eq_dependants vat_t.eq_pv_dependants[OF eq_pv], simp)
+ -- {* Since @{term "th'"} is running, by @{thm runing_preced_inversion},
+ its @{term cp}-value equals @{term "preced th s"},
+ which equals to @{term "?R"} by simplification: *}
+ also have "... = ?R"
+ thm runing_preced_inversion
+ using runing_preced_inversion[OF otherwise] by simp
+ finally show ?thesis .
+ qed
+ qed (auto simp: th'_in th_kept)
+ with `th' \<noteq> th` show ?thesis by simp
+ qed
+qed
+
+text {*
+ The following lemma is the extrapolation of @{thm eq_pv_blocked}.
+ It says if a thread, different from @{term th},
+ does not hold any resource at the very beginning,
+ it will keep hand-emptied in the future @{term "t@s"}.
+*}
+lemma eq_pv_persist: (* ddd *)
+ assumes neq_th': "th' \<noteq> th"
+ and eq_pv: "cntP s th' = cntV s th'"
+ shows "cntP (t@s) th' = cntV (t@s) th'"
+proof(induction rule:ind) -- {* The proof goes by induction. *}
+ -- {* The nontrivial case is for the @{term Cons}: *}
+ case (Cons e t)
+ -- {* All results derived so far hold for both @{term s} and @{term "t@s"}: *}
+ 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
+ interpret vat_es: valid_trace_e "t@s" e using Cons(1,2) by (unfold_locales, auto)
+ show ?case
+ proof -
+ -- {* It can be proved that @{term cntP}-value of @{term th'} does not change
+ by the happening of event @{term e}: *}
+ have "cntP ((e#t)@s) th' = cntP (t@s) th'"
+ proof(rule ccontr) -- {* Proof by contradiction. *}
+ -- {* Suppose @{term cntP}-value of @{term th'} is changed by @{term e}: *}
+ assume otherwise: "cntP ((e # t) @ s) th' \<noteq> cntP (t @ s) th'"
+ -- {* Then the actor of @{term e} must be @{term th'} and @{term e}
+ must be a @{term P}-event: *}
+ hence "isP e" "actor e = th'" by (auto simp:cntP_diff_inv)
+ with vat_es.actor_inv
+ -- {* According to @{thm vat_es.actor_inv}, @{term th'} must be running at
+ the moment @{term "t@s"}: *}
+ have "th' \<in> runing (t@s)" by (cases e, auto)
+ -- {* However, an application of @{thm eq_pv_blocked} to induction hypothesis
+ shows @{term th'} can not be running at moment @{term "t@s"}: *}
+ moreover have "th' \<notin> runing (t@s)"
+ using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] .
+ -- {* Contradiction is finally derived: *}
+ ultimately show False by simp
+ qed
+ -- {* It can also be proved that @{term cntV}-value of @{term th'} does not change
+ by the happening of event @{term e}: *}
+ -- {* The proof follows exactly the same pattern as the case for @{term cntP}-value: *}
+ moreover have "cntV ((e#t)@s) th' = cntV (t@s) th'"
+ proof(rule ccontr) -- {* Proof by contradiction. *}
+ assume otherwise: "cntV ((e # t) @ s) th' \<noteq> cntV (t @ s) th'"
+ hence "isV e" "actor e = th'" by (auto simp:cntV_diff_inv)
+ with vat_es.actor_inv
+ have "th' \<in> runing (t@s)" by (cases e, auto)
+ moreover have "th' \<notin> runing (t@s)"
+ using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] .
+ ultimately show False by simp
+ qed
+ -- {* Finally, it can be shown that the @{term cntP} and @{term cntV}
+ value for @{term th'} are still in balance, so @{term th'}
+ is still hand-emptied after the execution of event @{term e}: *}
+ ultimately show ?thesis using Cons(5) by metis
+ qed
+qed (auto simp:eq_pv)
+
+text {*
+ By combining @{thm eq_pv_blocked} and @{thm eq_pv_persist},
+ it can be derived easily that @{term th'} can not be running in the future:
+*}
+lemma eq_pv_blocked_persist:
+ assumes neq_th': "th' \<noteq> th"
+ and eq_pv: "cntP s th' = cntV s th'"
+ shows "th' \<notin> runing (t@s)"
+ using assms
+ by (simp add: eq_pv_blocked eq_pv_persist)
+
+text {*
+ The following lemma shows the blocking thread @{term th'}
+ must hold some resource in the very beginning.
+*}
+lemma runing_cntP_cntV_inv: (* ddd *)
+ assumes is_runing: "th' \<in> runing (t@s)"
+ and neq_th': "th' \<noteq> th"
+ shows "cntP s th' > cntV s th'"
+ using assms
+proof -
+ -- {* First, it can be shown that the number of @{term P} and
+ @{term V} operations can not be equal for thred @{term th'} *}
+ have "cntP s th' \<noteq> cntV s th'"
+ proof
+ -- {* The proof goes by contradiction, suppose otherwise: *}
+ assume otherwise: "cntP s th' = cntV s th'"
+ -- {* By applying @{thm eq_pv_blocked_persist} to this: *}
+ from eq_pv_blocked_persist[OF neq_th' otherwise]
+ -- {* we have that @{term th'} can not be running at moment @{term "t@s"}: *}
+ have "th' \<notin> runing (t@s)" .
+ -- {* This is obvious in contradiction with assumption @{thm is_runing} *}
+ thus False using is_runing by simp
+ qed
+ -- {* However, the number of @{term V} is always less or equal to @{term P}: *}
+ moreover have "cntV s th' \<le> cntP s th'" using vat_s.cnp_cnv_cncs by auto
+ -- {* Thesis is finally derived by combining the these two results: *}
+ ultimately show ?thesis by auto
+qed
+
+
+text {*
+ The following lemmas shows the blocking thread @{text th'} must be live
+ at the very beginning, i.e. the moment (or state) @{term s}.
+
+ The proof is a simple combination of the results above:
+*}
+lemma runing_threads_inv:
+ assumes runing': "th' \<in> runing (t@s)"
+ and neq_th': "th' \<noteq> th"
+ shows "th' \<in> threads s"
+proof(rule ccontr) -- {* Proof by contradiction: *}
+ assume otherwise: "th' \<notin> threads s"
+ have "th' \<notin> runing (t @ s)"
+ proof -
+ from vat_s.cnp_cnv_eq[OF otherwise]
+ have "cntP s th' = cntV s th'" .
+ from eq_pv_blocked_persist[OF neq_th' this]
+ show ?thesis .
+ qed
+ with runing' show False by simp
+qed
+
+text {*
+ The following lemma summarizes several foregoing
+ lemmas to give an overall picture of the blocking thread @{text "th'"}:
+*}
+lemma runing_inversion: (* ddd, one of the main lemmas to present *)
+ 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"
+proof -
+ from runing_threads_inv[OF assms]
+ show "th' \<in> threads s" .
+next
+ from runing_cntP_cntV_inv[OF runing' neq_th]
+ show "\<not>detached s th'" using vat_s.detached_eq by simp
+next
+ from runing_preced_inversion[OF runing']
+ show "cp (t@s) th' = preced th s" .
+qed
+
+section {* The existence of `blocking thread` *}
+
+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, the other main lemma to be presented: *)
+ 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.rg_RAG_threads vat_t.subtree_tRAG_thread)
+ next
+ show "Th th \<in> subtree (tRAG (t @ s)) (Th th')" using dp
+ by (unfold tRAG_subtree_eq, auto simp:subtree_def)
+ next
+ show "Max ((the_preced (t @ s) \<circ> the_thread) ` Th ` threads (t @ s)) =
+ (the_preced (t @ s) \<circ> the_thread) (Th th)" (is "Max ?L = _")
+ proof -
+ have "?L = the_preced (t @ s) ` threads (t @ s)"
+ by (unfold image_comp, rule image_cong, auto)
+ thus ?thesis using max_preced the_preced_def by auto
+ qed
+ qed
+ also have "... = ?R"
+ using th_cp_max th_cp_preced th_kept
+ the_preced_def vat_t.max_cp_readys_threads by auto
+ finally show ?thesis .
+ qed
+ -- {* Now, since @{term th'} holds the highest @{term cp}
+ and we have already show it is in @{term readys},
+ it is @{term runing} by definition. *}
+ with `th' \<in> readys (t@s)` show ?thesis by (simp add: runing_def)
+ qed
+ -- {* It is easy to show @{term th'} is an ancestor of @{term th}: *}
+ moreover have "Th th' \<in> ancestors (RAG (t @ s)) (Th th)"
+ using `(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+` by (auto simp:ancestors_def)
+ ultimately show ?thesis using that by metis
+qed
+
+text {*
+ Now it is easy to see there is always a thread to run by case analysis
+ on whether thread @{term th} is running: if the answer is Yes, the
+ the running thread is obviously @{term th} itself; otherwise, the running
+ thread is the @{text th'} given by lemma @{thm th_blockedE}.
+*}
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)")
+ case True thus ?thesis by auto
+next
+ case False
+ thus ?thesis using th_blockedE by auto
+qed
+
+end
+end
--- a/Correctness.thy Tue Jun 14 15:06:16 2016 +0100
+++ b/Correctness.thy Fri Jun 17 09:46:25 2016 +0100
@@ -558,7 +558,7 @@
of it contains only itself, so, its @{term cp}-value
equals its @{term preced}-value: *}
have "?L = cp (t@s) th'"
- by (unfold cp_eq cpreced_def eq_dependants vat_t.eq_pv_dependants[OF eq_pv], simp)
+ by (simp add: detached_cp_preced eq_pv vat_t.detached_intro)
-- {* Since @{term "th'"} is running, by @{thm running_preced_inversion},
its @{term cp}-value equals @{term "preced th s"},
which equals to @{term "?R"} by simplification: *}
--- a/ExtGG.thy Tue Jun 14 15:06:16 2016 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,702 +0,0 @@
-section {*
- This file contains lemmas used to guide the recalculation of current precedence
- after every system call (or system operation)
-*}
-theory Implementation
-imports PIPBasics
-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
-
-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_es.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_es.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_es.ancestors_headE[OF assms(1) start])
- case 1
- from x_u[folded this, unfolded eq_x] vat_es.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_es.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_es.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_es.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_es.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
-
-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_es.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
-
--- a/Implementation.thy Tue Jun 14 15:06:16 2016 +0100
+++ b/Implementation.thy Fri Jun 17 09:46:25 2016 +0100
@@ -154,7 +154,7 @@
proof -
from holding_th_cs_s
show ?thesis
- by (unfold s_RAG_def holding_eq, auto)
+ by (unfold s_RAG_def s_holding_abv, auto)
qed
lemma ancestors_cs:
@@ -187,7 +187,7 @@
lemma sub_RAGs':
"{(Cs cs, Th th), (Th taker, Cs cs)} \<subseteq> RAG s"
using waiting_taker holding_th_cs_s
- by (unfold s_RAG_def, fold waiting_eq holding_eq, auto)
+ by (unfold s_RAG_def, fold s_waiting_abv s_holding_abv, auto)
lemma ancestors_th':
"ancestors (RAG s) (Th taker) = {Th th, Cs cs}"
@@ -297,7 +297,7 @@
by (auto simp:ancestors_def)
from tranclD2[OF this]
obtain th' where "waiting s th' cs"
- by (auto simp:s_RAG_def waiting_eq)
+ by (auto simp:s_RAG_def s_waiting_abv)
with no_waiter_before
show ?thesis by simp
qed simp
@@ -390,7 +390,7 @@
lemma cs_held: "(Cs cs, Th holder) \<in> RAG s"
using holding_s_holder
- by (unfold s_RAG_def, fold holding_eq, auto)
+ by (unfold s_RAG_def, fold s_holding_abv, auto)
lemma tRAG_s:
"tRAG (e#s) = tRAG s \<union> {(Th th, Th holder)}"
@@ -662,7 +662,7 @@
assume "Th th \<in> Range (RAG s)"
then obtain cs where "holding s th cs"
by (simp add: holdents_RAG holdents_th_s)
- then show False by (unfold holding_eq, auto)
+ then show False by (unfold s_holding_abv, auto)
qed
moreover have "Th th \<notin> Domain (RAG s)"
proof
--- a/Journal/Paper.thy Tue Jun 14 15:06:16 2016 +0100
+++ b/Journal/Paper.thy Fri Jun 17 09:46:25 2016 +0100
@@ -33,18 +33,21 @@
vt ("valid'_state") and
Prc ("'(_, _')") and
holding_raw ("holds") and
- holding ("Holds") and
+ holding ("holds") and
waiting_raw ("waits") and
- waiting ("Waits") and
+ waiting ("waits") and
dependants_raw ("dependants") and
- dependants ("Dependants") and
+ dependants ("dependants") and
+ RAG_raw ("RAG") and
+ RAG ("RAG") and
Th ("T") and
Cs ("C") and
readys ("ready") and
preced ("prec") and
preceds ("precs") and
cpreced ("cprec") and
- cp ("cprec") and
+ wq_fun ("wq") and
+ cprec_fun ("cp") and
holdents ("resources") and
DUMMY ("\<^raw:\mbox{$\_\!\_$}>") and
cntP ("c\<^bsub>P\<^esub>") and
@@ -455,7 +458,7 @@
\noindent
Using @{term "holding_raw"} and @{term waiting_raw}, we can introduce \emph{Resource Allocation Graphs}
(RAG), which represent the dependencies between threads and resources.
- We represent RAGs as relations using pairs of the form
+ We choose to represent RAGs as relations using pairs of the form
\begin{isabelle}\ \ \ \ \ %%%
@{term "(Th th, Cs cs)"} \hspace{5mm}{\rm and}\hspace{5mm}
@@ -524,8 +527,6 @@
that the resource is locked. In this way we can always start at a thread waiting for a
resource and ``chase'' outgoing arrows leading to a single root of a tree.
-
-
The use of relations for representing RAGs allows us to conveniently define
the notion of the \emph{dependants} of a thread using the transitive closure
operation for relations, written ~@{term "trancl DUMMY"}. This gives
@@ -545,14 +546,14 @@
there is a circle of dependencies in a RAG, then clearly we have a
deadlock. Therefore when a thread requests a resource, we must
ensure that the resulting RAG is not circular. In practice, the
- programmer has to ensure this.
-
+ programmer has to ensure this. Our model will assume that critical
+ reseources can only be requested provided no circularity can arise.
Next we introduce the notion of the \emph{current precedence} of a thread @{text th} in a
state @{text s}. It is defined as
\begin{isabelle}\ \ \ \ \ %%%
- @{thm cpreced_def2}\hfill\numbered{cpreced}
+ @{thm cpreced_def}\hfill\numbered{cpreced}
\end{isabelle}
\noindent
@@ -568,13 +569,17 @@
lowered prematurely. We again introduce an abbreviation for current precedeces of
a set of threads, written @{term "cprecs wq s ths"}.
+ \begin{isabelle}\ \ \ \ \ %%%
+ @{thm cpreceds_def}
+ \end{isabelle}
+
The next function, called @{term schs}, defines the behaviour of the scheduler. It will be defined
by recursion on the state (a list of events); this function returns a \emph{schedule state}, which
we represent as a record consisting of two
functions:
\begin{isabelle}\ \ \ \ \ %%%
- @{text "\<lparr>wq_fun, cprec_fun\<rparr>"}
+ @{text "\<lparr>wq, cp\<rparr>"}
\end{isabelle}
\noindent
--- a/PIPBasics.thy Tue Jun 14 15:06:16 2016 +0100
+++ b/PIPBasics.thy Fri Jun 17 09:46:25 2016 +0100
@@ -1,5 +1,5 @@
theory PIPBasics
-imports PIPDefs
+imports PIPDefs RTree
begin
text {* (* ddd *)
@@ -147,7 +147,7 @@
obtain th' where "th' \<in> set (wq s cs)" "th' = hd (wq s cs)"
by (metis empty_iff hd_in_set list.set(1))
hence "holding s th' cs"
- by (unfold s_holding_def, fold wq_def, auto)
+ unfolding s_holding_def by auto
from that[OF this] show ?thesis .
qed
@@ -159,7 +159,7 @@
*}
lemma children_RAG_alt_def:
"children (RAG (s::state)) (Th th) = Cs ` {cs. holding s th cs}"
- by (unfold s_RAG_def, auto simp:children_def holding_eq)
+ by (unfold s_RAG_def, auto simp:children_def s_holding_abv)
text {*
The following two lemmas relate @{term holdents} and @{term cntCS}
@@ -279,8 +279,8 @@
lemma in_RAG_E:
assumes "(n1, n2) \<in> RAG (s::state)"
obtains (waiting) th cs where "n1 = Th th" "n2 = Cs cs" "waiting s th cs"
- | (holding) th cs where "n1 = Cs cs" "n2 = Th th" "holding s th cs"
- using assms[unfolded s_RAG_def, folded waiting_eq holding_eq]
+ | (holding) th cs where "n1 = Cs cs" "n2 = Th th" "holding s th cs"
+ using assms[unfolded s_RAG_def, folded s_waiting_abv s_holding_abv]
by auto
text {*
@@ -584,8 +584,8 @@
thus ?thesis by simp
qed
thus ?thesis
- by (metis (no_types, lifting) cp_eq cpreced_def eq_dependants
- f_image_eq the_preced_def)
+ by (metis (no_types, lifting) cp_eq cpreced_def2 f_image_eq
+ s_dependants_abv the_preced_def)
qed
text {*
@@ -625,7 +625,7 @@
from h1 have "cs' = cs" by simp
from assms(2) cs_in[unfolded this]
have "holding s th'' cs" "holding s th2 cs"
- by (unfold s_RAG_def, fold holding_eq, auto)
+ by (unfold s_RAG_def, fold s_holding_abv, auto)
from held_unique[OF this]
show ?thesis by simp
qed
@@ -990,9 +990,9 @@
obtain rest where eq_wq: "wq s cs = th#rest" by blast
with otherwise
have "holding s th cs"
- by (unfold s_holding_def, fold wq_def, simp)
+ unfolding s_holding_def by auto
hence cs_th_RAG: "(Cs cs, Th th) \<in> RAG s"
- by (unfold s_RAG_def, fold holding_eq, auto)
+ by (unfold s_RAG_def, fold s_holding_abv, auto)
from pip_e[unfolded is_p]
show False
proof(cases)
@@ -1033,8 +1033,8 @@
proof(cases)
case (thread_V)
from this(2) show ?thesis
- by (unfold rest_def s_holding_def, fold wq_def,
- metis empty_iff list.collapse list.set(1))
+ unfolding s_holding_def
+ by (metis empty_iff empty_set hd_Cons_tl rest_def)
qed
qed
@@ -1168,8 +1168,9 @@
proof -
from pip_e[unfolded is_exit]
show ?thesis
- by (cases, unfold holdents_def s_holding_def, fold wq_def,
- auto elim!:running_wqE)
+ apply(cases)
+ unfolding holdents_def s_holding_def
+ by (metis (mono_tags, lifting) empty_iff list.sel(1) mem_Collect_eq running_wqE)
qed
lemma wq_threads_kept:
@@ -1562,7 +1563,7 @@
proof -
from assms(1) have "wq (e#s) c = wq s c" by auto
from assms(2)[unfolded s_holding_def, folded wq_def,
- folded this, unfolded wq_def, folded s_holding_def]
+ folded this, folded s_holding_def]
show ?thesis .
qed
@@ -1624,7 +1625,7 @@
lemma holding_taker:
shows "holding (e#s) taker cs"
- by (unfold s_holding_def, fold wq_def, unfold wq_es_cs,
+ by (unfold s_holding_def, unfold wq_es_cs,
auto simp:neq_wq' taker_def)
lemma waiting_esI2:
@@ -1692,7 +1693,7 @@
case False
hence "wq (e#s) c = wq s c" by auto
from assms[unfolded s_holding_def, folded wq_def,
- unfolded this, unfolded wq_def, folded s_holding_def]
+ unfolded this, folded s_holding_def]
have "holding s t c" .
from that(2)[OF False this] show ?thesis .
qed
@@ -1795,7 +1796,7 @@
case False
hence "wq (e#s) c = wq s c" by auto
from assms[unfolded s_holding_def, folded wq_def,
- unfolded this, unfolded wq_def, folded s_holding_def]
+ unfolded this, folded s_holding_def]
have "holding s t c" .
from that[OF False this] show ?thesis .
qed
@@ -1829,13 +1830,13 @@
with waiting(1,2)
show ?thesis
by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def,
- fold waiting_eq, auto)
+ fold s_waiting_abv, auto)
next
case 2
with waiting(1,2)
show ?thesis
by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def,
- fold waiting_eq, auto)
+ fold s_waiting_abv, auto)
qed
next
case True
@@ -1848,7 +1849,7 @@
with waiting(1,2)
show ?thesis
by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def,
- fold waiting_eq, auto)
+ fold s_waiting_abv, auto)
qed
qed
next
@@ -1865,13 +1866,13 @@
with holding(1,2)
show ?thesis
by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def,
- fold waiting_eq, auto)
+ fold s_waiting_abv, auto)
next
case 2
with holding(1,2)
show ?thesis
by (unfold h_n.waiting_set_eq h_n.holding_set_eq s_RAG_def,
- fold holding_eq, auto)
+ fold s_holding_abv, auto)
qed
next
case True
@@ -1884,7 +1885,7 @@
with holding(1,2)
show ?thesis
by (unfold h_e.waiting_set_eq h_e.holding_set_eq s_RAG_def,
- fold holding_eq, auto)
+ fold s_holding_abv, auto)
qed
qed
qed
@@ -1906,7 +1907,7 @@
assume "n2 = Th h_n.taker \<and> n1 = Cs cs"
with h_n.holding_taker
show ?thesis
- by (unfold s_RAG_def, fold holding_eq, auto)
+ by (unfold s_RAG_def, fold s_holding_abv, auto)
next
assume h: "(n1, n2) \<in> RAG s \<and>
(n1 \<noteq> Cs cs \<or> n2 \<noteq> Th th) \<and> (n1 \<noteq> Th h_n.taker \<or> n2 \<noteq> Cs cs)"
@@ -1935,7 +1936,7 @@
qed
qed
thus ?thesis using waiting(1,2)
- by (unfold s_RAG_def, fold waiting_eq, auto)
+ by (unfold s_RAG_def, fold s_waiting_abv, auto)
next
case (holding th' cs')
from h this(1,2)
@@ -1951,7 +1952,7 @@
show ?thesis .
qed
thus ?thesis using holding(1,2)
- by (unfold s_RAG_def, fold holding_eq, auto)
+ by (unfold s_RAG_def, fold s_holding_abv, auto)
qed
qed
next
@@ -1967,7 +1968,7 @@
case (waiting th' cs')
from h_e.waiting_esI2[OF this(3)]
show ?thesis using waiting(1,2)
- by (unfold s_RAG_def, fold waiting_eq, auto)
+ by (unfold s_RAG_def, fold s_waiting_abv, auto)
next
case (holding th' cs')
with h_s(2)
@@ -1977,12 +1978,12 @@
assume neq_cs: "cs' \<noteq> cs"
from holding_esI2[OF this holding(3)]
show ?thesis using holding(1,2)
- by (unfold s_RAG_def, fold holding_eq, auto)
+ by (unfold s_RAG_def, fold s_holding_abv, auto)
next
assume "th' \<noteq> th"
from holding_esI1[OF holding(3) this]
show ?thesis using holding(1,2)
- by (unfold s_RAG_def, fold holding_eq, auto)
+ by (unfold s_RAG_def, fold s_holding_abv, auto)
qed
qed
qed
@@ -2006,7 +2007,7 @@
proof(cases "cs' = cs")
case False
hence "wq (e#s) cs' = wq s cs'" by simp
- with assms show ?thesis unfolding holding_raw_def holding_eq by auto
+ with assms show ?thesis unfolding holding_raw_def s_holding_abv by auto
next
case True
from assms[unfolded s_holding_def, folded wq_def]
@@ -2015,7 +2016,7 @@
hence "wq (e#s) cs' = th'#(rest@[th])"
by (simp add: True wq_es_cs)
thus ?thesis
- by (simp add: holding_raw_def holding_eq)
+ by (simp add: holding_raw_def s_holding_abv)
qed
end
@@ -2038,11 +2039,11 @@
proof -
from wq_es_cs'
have "th \<in> set (wq (e#s) cs)" "th = hd (wq (e#s) cs)" by auto
- thus ?thesis unfolding holding_raw_def holding_eq by blast
+ thus ?thesis unfolding holding_raw_def s_holding_abv by blast
qed
lemma RAG_edge: "(Cs cs, Th th) \<in> RAG (e#s)"
- by (unfold s_RAG_def, fold holding_eq, insert holding_es_th_cs, auto)
+ by (unfold s_RAG_def, fold s_holding_abv, insert holding_es_th_cs, auto)
lemma waiting_esE:
assumes "waiting (e#s) th' cs'"
@@ -2063,7 +2064,7 @@
next
case False
have "holding s th' cs'" using assms
- using False unfolding holding_raw_def holding_eq by auto
+ using False unfolding holding_raw_def s_holding_abv by auto
from that(1)[OF False this] show ?thesis .
qed
@@ -2079,7 +2080,7 @@
proof(cases rule:waiting_esE)
case 1
thus ?thesis using waiting(1,2)
- by (unfold s_RAG_def, fold waiting_eq, auto)
+ by (unfold s_RAG_def, fold s_waiting_abv, auto)
qed
next
case (holding th' cs')
@@ -2088,7 +2089,7 @@
proof(cases rule:holding_esE)
case 1
with holding(1,2)
- show ?thesis by (unfold s_RAG_def, fold holding_eq, auto)
+ show ?thesis by (unfold s_RAG_def, fold s_holding_abv, auto)
next
case 2
with holding(1,2) show ?thesis by auto
@@ -2106,18 +2107,18 @@
case (waiting th' cs')
from waiting_kept[OF this(3)]
show ?thesis using waiting(1,2)
- by (unfold s_RAG_def, fold waiting_eq, auto)
+ by (unfold s_RAG_def, fold s_waiting_abv, auto)
next
case (holding th' cs')
from holding_kept[OF this(3)]
show ?thesis using holding(1,2)
- by (unfold s_RAG_def, fold holding_eq, auto)
+ by (unfold s_RAG_def, fold s_holding_abv, auto)
qed
next
assume "n1 = Cs cs \<and> n2 = Th th"
with holding_es_th_cs
show ?thesis
- by (unfold s_RAG_def, fold holding_eq, auto)
+ by (unfold s_RAG_def, fold s_holding_abv, auto)
qed
qed
@@ -2133,11 +2134,12 @@
by (simp add: wq_es_cs wq_s_cs)
lemma waiting_es_th_cs: "waiting (e#s) th cs"
- using th_not_in_wq waiting_eq wq_es_cs' wq_s_cs
- by (simp add: s_waiting_def wq_def wq_es_cs)
+ using th_not_in_wq s_waiting_abv wq_es_cs' wq_s_cs
+ using Un_iff list.sel(1) list.set_intros(1) s_waiting_def
+ set_append wq_def wq_es_cs by auto
lemma RAG_edge: "(Th th, Cs cs) \<in> RAG (e#s)"
- by (unfold s_RAG_def, fold waiting_eq, insert waiting_es_th_cs, auto)
+ by (unfold s_RAG_def, fold s_waiting_abv, insert waiting_es_th_cs, auto)
lemma holding_esE:
assumes "holding (e#s) th' cs'"
@@ -2187,7 +2189,7 @@
proof(cases rule:waiting_esE)
case 1
thus ?thesis using waiting(1,2)
- by (unfold s_RAG_def, fold waiting_eq, auto)
+ by (unfold s_RAG_def, fold s_waiting_abv, auto)
next
case 2
thus ?thesis using waiting(1,2) by auto
@@ -2199,7 +2201,7 @@
proof(cases rule:holding_esE)
case 1
with holding(1,2)
- show ?thesis by (unfold s_RAG_def, fold holding_eq, auto)
+ show ?thesis by (unfold s_RAG_def, fold s_holding_abv, auto)
qed
qed
next
@@ -2214,12 +2216,12 @@
case (waiting th' cs')
from waiting_kept[OF this(3)]
show ?thesis using waiting(1,2)
- by (unfold s_RAG_def, fold waiting_eq, auto)
+ by (unfold s_RAG_def, fold s_waiting_abv, auto)
next
case (holding th' cs')
from holding_kept[OF this(3)]
show ?thesis using holding(1,2)
- by (unfold s_RAG_def, fold holding_eq, auto)
+ by (unfold s_RAG_def, fold s_holding_abv, auto)
qed
next
assume "n1 = Th th \<and> n2 = Cs cs"
@@ -2620,7 +2622,7 @@
"(Th taker, Cs cs') \<in> RAG s"
by (unfold s_RAG_def, auto)
from this(2) have "waiting s taker cs'"
- by (unfold s_RAG_def, fold waiting_eq, auto)
+ by (unfold s_RAG_def, fold s_waiting_abv, auto)
from waiting_unique[OF this waiting_taker]
have "cs' = cs" .
from h(1)[unfolded this] show False by auto
@@ -2655,7 +2657,7 @@
obtain cs' where h: "(Th th, Cs cs') \<in> RAG s"
by (unfold s_RAG_def, auto)
hence "waiting s th cs'"
- by (unfold s_RAG_def, fold waiting_eq, auto)
+ by (unfold s_RAG_def, fold s_waiting_abv, auto)
with th_not_waiting show False by auto
qed
ultimately show ?thesis by auto
@@ -2784,7 +2786,7 @@
begin
lemma unique_RAG: "\<lbrakk>(n, n1) \<in> RAG s; (n, n2) \<in> RAG s\<rbrakk> \<Longrightarrow> n1 = n2"
- apply(unfold s_RAG_def, auto, fold waiting_eq holding_eq)
+ apply(unfold s_RAG_def, auto, fold s_waiting_abv s_holding_abv)
by(auto elim:waiting_unique held_unique)
lemma sgv_RAG: "single_valued (RAG s)"
@@ -2962,11 +2964,11 @@
obtain n where "(n, b) \<in> RAG s" by auto
from this[unfolded Cs]
obtain th1 where "waiting s th1 cs"
- by (unfold s_RAG_def, fold waiting_eq, auto)
+ by (unfold s_RAG_def, fold s_waiting_abv, auto)
from waiting_holding[OF this]
obtain th2 where "holding s th2 cs" .
hence "(Cs cs, Th th2) \<in> RAG s"
- by (unfold s_RAG_def, fold holding_eq, auto)
+ by (unfold s_RAG_def, fold s_holding_abv, auto)
with h_b(2)[unfolded Cs, rule_format]
have False by auto
thus ?thesis by auto
@@ -2975,7 +2977,7 @@
proof -
from h_b(2)[unfolded eq_b]
have "\<forall>cs. \<not> waiting s th' cs"
- by (unfold s_RAG_def, fold waiting_eq, auto)
+ by (unfold s_RAG_def, fold s_waiting_abv, auto)
moreover have "th' \<in> threads s"
proof(rule rg_RAG_threads)
from tranclD[OF h_b(1), unfolded eq_b]
@@ -3123,7 +3125,7 @@
have "(Th th1, Th th2) \<in> (RAG s)\<^sup>+" .
from tranclD[OF this]
obtain cs where "waiting s th1 cs"
- by (unfold s_RAG_def, fold waiting_eq, auto)
+ by (unfold s_RAG_def, fold s_waiting_abv, auto)
with running_1 show False
by (unfold running_def readys_def, auto)
qed
@@ -3139,7 +3141,7 @@
have "(Th th2, Th th1) \<in> (RAG s)\<^sup>+" .
from tranclD[OF this]
obtain cs where "waiting s th2 cs"
- by (unfold s_RAG_def, fold waiting_eq, auto)
+ by (unfold s_RAG_def, fold s_waiting_abv, auto)
with running_2 show False
by (unfold running_def readys_def, auto)
qed
@@ -3232,7 +3234,7 @@
obtain z where "(Th th1, z) \<in> RAG s" by auto
from this[unfolded s_RAG_def, folded wq_def]
obtain cs' where "waiting s th1 cs'"
- by (auto simp:waiting_eq)
+ by (auto simp:s_waiting_abv)
with assms(1) show False by (auto simp:readys_def)
qed
next
@@ -3251,7 +3253,7 @@
obtain z where "(Th th2, z) \<in> RAG s" by auto
from this[unfolded s_RAG_def, folded wq_def]
obtain cs' where "waiting s th2 cs'"
- by (auto simp:waiting_eq)
+ by (auto simp:s_waiting_abv)
with assms(2) show False by (auto simp:readys_def)
qed
qed
@@ -3425,10 +3427,10 @@
begin
lemma holding_s_holder: "holding s holder cs"
- by (unfold s_holding_def, fold wq_def, unfold wq_s_cs, auto)
+ by (unfold s_holding_def, unfold wq_s_cs, auto)
lemma holding_es_holder: "holding (e#s) holder cs"
- by (unfold s_holding_def, fold wq_def, unfold wq_es_cs wq_s_cs, auto)
+ by (unfold s_holding_def, unfold wq_es_cs wq_s_cs, auto)
lemma holdents_es:
shows "holdents (e#s) th' = holdents s th'" (is "?L = ?R")
@@ -3448,7 +3450,7 @@
hence "wq (e#s) cs' = wq s cs'" by simp
from h[unfolded s_holding_def, folded wq_def, unfolded this]
show ?thesis
- by (unfold s_holding_def, fold wq_def, auto)
+ by (unfold s_holding_def, auto)
qed
hence "cs' \<in> ?R" by (auto simp:holdents_def)
} moreover {
@@ -3467,7 +3469,7 @@
hence "wq s cs' = wq (e#s) cs'" by simp
from h[unfolded s_holding_def, folded wq_def, unfolded this]
show ?thesis
- by (unfold s_holding_def, fold wq_def, auto)
+ by (unfold s_holding_def, auto)
qed
hence "cs' \<in> ?L" by (auto simp:holdents_def)
} ultimately show ?thesis by auto
@@ -3598,7 +3600,7 @@
from h_e[unfolded s_holding_def, folded wq_def, unfolded wq_neq_simp[OF this]]
have "th' \<in> set (wq s cs') \<and> th' = hd (wq s cs')" .
hence "cs' \<in> ?R"
- by (unfold holdents_def s_holding_def, fold wq_def, auto)
+ by (unfold holdents_def s_holding_def, auto)
} moreover {
fix cs'
assume "cs' \<in> ?R"
@@ -3738,7 +3740,7 @@
lemma holding_th_cs_s:
"holding s th cs"
- by (unfold s_holding_def, fold wq_def, unfold wq_s_cs, auto)
+ by (unfold s_holding_def, unfold wq_s_cs, auto)
lemma th_ready_s [simp]: "th \<in> readys s"
using running_th_s
@@ -3931,7 +3933,7 @@
from h have "holding (e#s) th' cs'" by (auto simp:holdents_def)
from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq]
show ?thesis
- by (unfold holdents_def s_holding_def, fold wq_def, auto)
+ by (unfold holdents_def s_holding_def, auto)
next
case True
from h[unfolded this]
@@ -3950,7 +3952,7 @@
from h have "holding s th' cs'" by (auto simp:holdents_def)
from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq]
show ?thesis
- by (unfold holdents_def s_holding_def, fold wq_def, insert eq_wq, simp)
+ by (unfold holdents_def s_holding_def, insert eq_wq, simp)
next
case True
from h[unfolded this]
@@ -4107,7 +4109,7 @@
from h have "holding (e#s) th' cs'" by (auto simp:holdents_def)
from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq]
show ?thesis
- by (unfold holdents_def s_holding_def, fold wq_def, auto)
+ by (unfold holdents_def s_holding_def, auto)
next
case True
from h[unfolded this]
@@ -4126,7 +4128,7 @@
from h have "holding s th' cs'" by (auto simp:holdents_def)
from this[unfolded s_holding_def, folded wq_def, unfolded eq_wq]
show ?thesis
- by (unfold holdents_def s_holding_def, fold wq_def, insert eq_wq, simp)
+ by (unfold holdents_def s_holding_def, insert eq_wq, simp)
next
case True
from h[unfolded this]
@@ -4315,14 +4317,12 @@
{ fix cs'
assume h: "cs' \<in> ?L"
hence "cs' \<in> ?R"
- by (unfold holdents_def s_holding_def, fold wq_def,
- unfold wq_kept, auto)
+ by (unfold holdents_def s_holding_def, unfold wq_kept, auto)
} moreover {
fix cs'
assume h: "cs' \<in> ?R"
hence "cs' \<in> ?L"
- by (unfold holdents_def s_holding_def, fold wq_def,
- unfold wq_kept, auto)
+ by (unfold holdents_def s_holding_def, unfold wq_kept, auto)
} ultimately show ?thesis by auto
qed
@@ -4432,7 +4432,7 @@
assume "holding (e # s) th cs'"
from this[unfolded s_holding_def, folded wq_def, unfolded wq_kept]
have "holding s th cs'"
- by (unfold s_holding_def, fold wq_def, auto)
+ by (unfold s_holding_def, auto)
with not_holding_th_s
show False by simp
qed
@@ -4462,14 +4462,12 @@
{ fix cs'
assume h: "cs' \<in> ?L"
hence "cs' \<in> ?R"
- by (unfold holdents_def s_holding_def, fold wq_def,
- unfold wq_kept, auto)
+ by (unfold holdents_def s_holding_def, unfold wq_kept, auto)
} moreover {
fix cs'
assume h: "cs' \<in> ?R"
hence "cs' \<in> ?L"
- by (unfold holdents_def s_holding_def, fold wq_def,
- unfold wq_kept, auto)
+ by (unfold holdents_def s_holding_def, unfold wq_kept, auto)
} ultimately show ?thesis by auto
qed
@@ -4567,14 +4565,12 @@
{ fix cs'
assume h: "cs' \<in> ?L"
hence "cs' \<in> ?R"
- by (unfold holdents_def s_holding_def, fold wq_def,
- unfold wq_kept, auto)
+ by (unfold holdents_def s_holding_def, unfold wq_kept, auto)
} moreover {
fix cs'
assume h: "cs' \<in> ?R"
hence "cs' \<in> ?L"
- by (unfold holdents_def s_holding_def, fold wq_def,
- unfold wq_kept, auto)
+ by (unfold holdents_def s_holding_def, unfold wq_kept, auto)
} ultimately show ?thesis by auto
qed
@@ -4639,8 +4635,8 @@
proof(induct rule:ind)
case Nil
thus ?case
- by (unfold cntP_def cntV_def pvD_def cntCS_def holdents_def
- s_holding_def, simp)
+ unfolding cntP_def cntV_def pvD_def cntCS_def holdents_def s_holding_def
+ by(simp add: wq_def)
next
case (Cons s e)
interpret vt_e: valid_trace_e s e using Cons by simp
@@ -4772,7 +4768,7 @@
lemma count_eq_tRAG_plus:
assumes "cntP s th = cntV s th"
shows "{th'. (Th th', Th th) \<in> (tRAG s)^+} = {}"
- using assms eq_pv_dependants dependants_alt_def eq_dependants by auto
+ using assms count_eq_RAG_plus dependants_alt_def s_dependants_def by blast
lemma count_eq_tRAG_plus_Th:
assumes "cntP s th = cntV s th"
@@ -4880,7 +4876,7 @@
with dtc
have "th \<in> readys s"
by (unfold readys_def detached_def Field_def Domain_def Range_def,
- auto simp:waiting_eq s_RAG_def)
+ auto simp:s_waiting_abv s_RAG_def)
with cncs_z show ?thesis using cnp_cnv_cncs by (simp add:pvD_def)
next
case False
--- a/PIPDefs.thy Tue Jun 14 15:06:16 2016 +0100
+++ b/PIPDefs.thy Fri Jun 17 09:46:25 2016 +0100
@@ -1,6 +1,6 @@
(*<*)
theory PIPDefs
-imports Precedence_ord RTree Max
+imports Precedence_ord Max
begin
(*>*)
@@ -8,8 +8,8 @@
text {*
- In this section, the formal model of Priority Inheritance Protocol (PIP)
- is presented. The model is based on Paulson's inductive protocol
+ In this chapter, the formal model of the Priority Inheritance Protocol
+ (PIP) is presented. The model is based on Paulson's inductive protocol
verification method, where the state of the system is modelled as a list
of events (trace) happened so far with the latest event put at the head.
*}
@@ -18,12 +18,12 @@
To define events, the identifiers of {\em threads}, {\em priority} and
{\em critical resources } (abbreviated as @{text "cs"}) need to be
- represented. All three are represetned using standard Isabelle/HOL type
+ represented. All three are represented using standard Isabelle/HOL type
@{typ "nat"}: *}
-type_synonym thread = nat -- {* Type for thread identifiers. *}
+type_synonym thread = nat -- {* Type for thread identifiers. *}
type_synonym priority = nat -- {* Type for priorities. *}
-type_synonym cs = nat -- {* Type for critical sections (or critical resources). *}
+type_synonym cs = nat -- {* Type for critical sections (or critical resources). *}
text {*
@@ -38,33 +38,13 @@
| V thread cs -- {* Thread @{text "thread"} releasing critical resource @{text "cs"}. *}
| Set thread priority -- {* Thread @{text "thread"} resets its priority to @{text "priority"}. *}
-fun actor where
- "actor (Exit th) = th" |
- "actor (P th cs) = th" |
- "actor (V th cs) = th" |
- "actor (Set th pty) = th" |
- "actor (Create th prio) = th"
--- {* The actions of a set of threads *}
-definition "actions_of ths s = filter (\<lambda> e. actor e \<in> ths) s"
-
-fun isCreate :: "event \<Rightarrow> bool" where
- "isCreate (Create th pty) = True" |
- "isCreate _ = False"
-
-fun isP :: "event \<Rightarrow> bool" where
- "isP (P th cs) = True" |
- "isP _ = False"
-
-fun isV :: "event \<Rightarrow> bool" where
- "isV (V th cs) = True" |
- "isV _ = False"
text {*
- As mentioned earlier, in Paulson's inductive method, the states of system
- are represented as lists of events, which is defined by the following type
- @{text "state"}: *}
+ As mentioned earlier, in Paulson's inductive method, the states of the
+ system are represented as lists of events, which is defined by the
+ following type @{text "state"}: *}
type_synonym state = "event list"
@@ -91,7 +71,7 @@
function}s which forms the very basis of Paulson's inductive protocol
verification method. Each observation function {\em observes} one
particular aspect (or attribute) of the system. For example, the attribute
- observed by @{text "threads s"} is the set of threads living in state
+ observed by @{text "threads s"} is the set of threads being live in state
@{text "s"}. The protocol being modelled The decision made the protocol
being modelled is based on the {\em observation}s returned by {\em
observation function}s. Since {\observation function}s forms the very
@@ -102,11 +82,10 @@
text {*
- \noindent Observation @{text "priority th s"} is the {\em original
- priority} of thread @{text "th"} in state @{text "s"}. The {\em original
- priority} is the priority assigned to a thread when it is created or when
- it is reset by system call (represented by event @{text "Set thread
- priority"}). *}
+ Observation @{text "priority th s"} is the {\em original priority} of
+ thread @{text "th"} in state @{text "s"}. The {\em original priority} is
+ the priority assigned to a thread when it is created or when it is reset
+ by system call (represented by event @{text "Set thread priority"}). *}
fun priority :: "thread \<Rightarrow> state \<Rightarrow> priority"
where
@@ -209,10 +188,10 @@
text {*
- \begin{minipage}{0.9\textwidth} The following @{text "dependants wq th"}
- represents the set of threads which are RAGing on thread @{text "th"} in
- Resource Allocation Graph @{text "RAG wq"}. Here, "RAGing" means waiting
- directly or indirectly on the critical resource. \end{minipage} *}
+ \noindent The following @{text "dependants wq th"} represents the set of
+ threads which are waiting on thread @{text "th"} in Resource Allocation
+ Graph @{text "RAG wq"}. Here, "waiting" means waiting directly or
+ indirectly on the critical resource. *}
definition
dependants_raw :: "(cs \<Rightarrow> thread list) \<Rightarrow> thread \<Rightarrow> thread set"
@@ -233,7 +212,9 @@
definition
cpreced :: "(cs \<Rightarrow> thread list) \<Rightarrow> state \<Rightarrow> thread \<Rightarrow> precedence"
where
- "cpreced wq s th = Max ((\<lambda>th'. preced th' s) ` ({th} \<union> dependants_raw wq th))"
+ "cpreced wq s th \<equiv> Max ({preced th s} \<union> preceds (dependants_raw wq th) s)"
+
+
text {*
@@ -245,12 +226,18 @@
from. *}
lemma cpreced_def2:
- "cpreced wq s th \<equiv> Max ({preced th s} \<union> preceds (dependants_raw wq th) s)"
+ "cpreced wq s th \<equiv> Max ((\<lambda>th'. preced th' s) ` ({th} \<union> dependants_raw wq th))"
unfolding cpreced_def image_def preceds_def
apply(rule eq_reflection)
apply(rule_tac f="Max" in arg_cong)
by (auto)
+definition
+ cpreceds :: "(cs \<Rightarrow> thread list) \<Rightarrow> state \<Rightarrow> thread set \<Rightarrow> precedence set"
+ where
+ "cpreceds wq s ths \<equiv> {cpreced wq s th | th. th \<in> ths}"
+
+
text {*
\noindent Assuming @{text "qs"} be the waiting queue of a critical
@@ -392,7 +379,7 @@
apply(rule ext)
apply(simp add: cpreced_def)
apply(simp add: dependants_raw_def RAG_raw_def waiting_raw_def holding_raw_def)
-apply(simp add: preced_def)
+apply(simp add: preced_def preceds_def)
done
text {*
@@ -420,37 +407,19 @@
definition
s_holding_abv:
- "holding (s::state) \<equiv> holding_raw (wq_fun (schs s))"
+ "holding (s::state) \<equiv> holding_raw (wq s)"
definition
s_waiting_abv:
- "waiting (s::state) \<equiv> waiting_raw (wq_fun (schs s))"
+ "waiting (s::state) \<equiv> waiting_raw (wq s)"
definition
s_RAG_abv:
- "RAG (s::state) \<equiv> RAG_raw (wq_fun (schs s))"
+ "RAG (s::state) \<equiv> RAG_raw (wq s)"
definition
s_dependants_abv:
- "dependants (s::state) \<equiv> dependants_raw (wq_fun (schs s))"
-
-text {*
-
- The following four lemmas relate the @{term wq} and non-@{term wq}
- versions of @{term waiting}, @{term holding}, @{term dependants} and
- @{term cp}. *}
-
-lemma waiting_eq:
- shows "waiting s th cs = waiting_raw (wq s) th cs"
- by (simp add: s_waiting_abv wq_def)
-
-lemma holding_eq:
- shows "holding s th cs = holding_raw (wq s) th cs"
- by (simp add: s_holding_abv wq_def)
-
-lemma eq_dependants:
- shows "dependants_raw (wq s) = dependants s"
- by (simp add: s_dependants_abv wq_def)
+ "dependants (s::state) \<equiv> dependants_raw (wq s)"
lemma cp_eq:
shows "cp s th = cpreced (wq s) s th"
@@ -472,8 +441,8 @@
lemma
s_holding_def:
- "holding (s::state) th cs \<equiv> (th \<in> set (wq_fun (schs s) cs) \<and> th = hd (wq_fun (schs s) cs))"
- by (auto simp:s_holding_abv wq_def holding_raw_def)
+ "holding (s::state) th cs \<equiv> (th \<in> set (wq s cs) \<and> th = hd (wq s cs))"
+ by(simp add: s_holding_abv holding_raw_def)
lemma s_waiting_def:
"waiting (s::state) th cs \<equiv> (th \<in> set (wq_fun (schs s) cs) \<and> th \<noteq> hd (wq_fun (schs s) cs))"
@@ -721,14 +690,37 @@
text {* The following lemma splits @{term "RAG"} graph into the above two sub-graphs. *}
lemma RAG_split: "RAG s = (wRAG s \<union> hRAG s)"
- by (unfold s_RAG_abv wRAG_def hRAG_def s_waiting_abv
- s_holding_abv RAG_raw_def, auto)
+using hRAG_def s_RAG_def s_holding_abv s_waiting_abv wRAG_def wq_def by auto
lemma tRAG_alt_def:
"tRAG s = {(Th th1, Th th2) | th1 th2.
\<exists> cs. (Th th1, Cs cs) \<in> RAG s \<and> (Cs cs, Th th2) \<in> RAG s}"
by (auto simp:tRAG_def RAG_split wRAG_def hRAG_def)
+
+fun actor where
+ "actor (Exit th) = th" |
+ "actor (P th cs) = th" |
+ "actor (V th cs) = th" |
+ "actor (Set th pty) = th" |
+ "actor (Create th prio) = th"
+
+-- {* The actions of a set of threads *}
+definition "actions_of ths s = filter (\<lambda> e. actor e \<in> ths) s"
+
+fun isCreate :: "event \<Rightarrow> bool" where
+ "isCreate (Create th pty) = True" |
+ "isCreate _ = False"
+
+fun isP :: "event \<Rightarrow> bool" where
+ "isP (P th cs) = True" |
+ "isP _ = False"
+
+fun isV :: "event \<Rightarrow> bool" where
+ "isV (V th cs) = True" |
+ "isV _ = False"
+
+
(*<*)
end
--- a/PrioG.thy Tue Jun 14 15:06:16 2016 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,797 +0,0 @@
-theory Correctness
-imports PIPBasics
-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
-
-lemma highest_cp_preced: "cp s th = Max (the_preced s ` threads s)"
- using eq_cp_s_th highest max_cp_eq the_preced_def by presburger
-
-
-lemma highest_preced_thread: "preced th s = Max (the_preced s ` threads s)"
- by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
-
-lemma highest': "cp s th = Max (cp s ` threads s)"
- by (simp add: eq_cp_s_th highest)
-
-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 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[simp]: "Max (cp (t@s) ` threads (t@s)) = cp (t@s) th"
- using max_cp_eq th_cp_max_preced the_preced_def vt_t by presburger
-
-lemma [simp]: "Max (cp (t@s) ` threads (t@s)) = Max (the_preced (t@s) ` threads (t@s))"
- by (simp add: th_cp_max_preced)
-
-lemma [simp]: "Max (the_preced (t@s) ` threads (t@s)) = the_preced (t@s) th"
- using max_kept th_kept the_preced_def by auto
-
-lemma [simp]: "the_preced (t@s) th = preced th (t@s)"
- using the_preced_def by auto
-
-lemma [simp]: "preced th (t@s) = preced th s"
- by (simp add: th_kept)
-
-lemma [simp]: "cp s th = preced th s"
- by (simp add: eq_cp_s_th)
-
-lemma th_cp_preced [simp]: "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)
-
-section {* The `blocking thread` *}
-
-text {*
- The purpose of PIP is to ensure that the most
- urgent thread @{term th} is not blocked unreasonably.
- Therefore, a clear picture of the blocking thread is essential
- to assure people that the purpose is fulfilled.
-
- In this section, we are going to derive a series of lemmas
- with finally give rise to a picture of the blocking thread.
-
- By `blocking thread`, we mean a thread in running state but
- different from thread @{term th}.
-*}
-
-text {*
- The following lemmas shows that the @{term cp}-value
- of the blocking thread @{text th'} equals to the highest
- precedence in the whole system.
-*}
-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 following lemma shows how the counters for @{term "P"} and
- @{term "V"} operations relate to the running threads in the states
- @{term s} and @{term "t @ s"}. The lemma shows that if a thread's
- @{term "P"}-count equals its @{term "V"}-count (which means it no
- longer has any resource in its possession), it cannot be a running
- thread.
-
- The proof is by contraction with the assumption @{text "th' \<noteq> th"}.
- The key is the use of @{thm eq_pv_dependants} to derive the
- emptiness of @{text th'}s @{term dependants}-set from the balance of
- its @{term P} and @{term V} counts. From this, it can be shown
- @{text th'}s @{term cp}-value equals to its own precedence.
-
- On the other hand, since @{text th'} is running, by @{thm
- runing_preced_inversion}, its @{term cp}-value equals to the
- precedence of @{term th}.
-
- Combining the above two resukts we have that @{text th'} and @{term
- th} have the same precedence. By uniqueness of precedences, we have
- @{text "th' = th"}, which is in contradiction with the assumption
- @{text "th' \<noteq> th"}.
-
-*}
-
-lemma eq_pv_blocked: (* ddd *)
- assumes 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'_in: "th' \<in> threads (t@s)"
- using otherwise readys_threads runing_def by auto
- have "th' = th"
- proof(rule preced_unique)
- -- {* The proof goes like this:
- it is first shown that the @{term preced}-value of @{term th'}
- equals to that of @{term th}, then by uniqueness
- of @{term preced}-values (given by lemma @{thm preced_unique}),
- @{term th'} equals to @{term th}: *}
- show "preced th' (t @ s) = preced th (t @ s)" (is "?L = ?R")
- proof -
- -- {* Since the counts of @{term th'} are balanced, the subtree
- of it contains only itself, so, its @{term cp}-value
- equals its @{term preced}-value: *}
- have "?L = cp (t@s) th'"
- by (unfold cp_eq_cpreced cpreced_def eq_dependants vat_t.eq_pv_dependants[OF eq_pv], simp)
- -- {* Since @{term "th'"} is running, by @{thm runing_preced_inversion},
- its @{term cp}-value equals @{term "preced th s"},
- which equals to @{term "?R"} by simplification: *}
- also have "... = ?R"
- thm runing_preced_inversion
- using runing_preced_inversion[OF otherwise] by simp
- finally show ?thesis .
- qed
- qed (auto simp: th'_in th_kept)
- with `th' \<noteq> th` show ?thesis by simp
- qed
-qed
-
-text {*
- The following lemma is the extrapolation of @{thm eq_pv_blocked}.
- It says if a thread, different from @{term th},
- does not hold any resource at the very beginning,
- it will keep hand-emptied in the future @{term "t@s"}.
-*}
-lemma eq_pv_persist: (* ddd *)
- assumes neq_th': "th' \<noteq> th"
- and eq_pv: "cntP s th' = cntV s th'"
- shows "cntP (t@s) th' = cntV (t@s) th'"
-proof(induction rule:ind) -- {* The proof goes by induction. *}
- -- {* The nontrivial case is for the @{term Cons}: *}
- case (Cons e t)
- -- {* All results derived so far hold for both @{term s} and @{term "t@s"}: *}
- 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
- interpret vat_es: valid_trace_e "t@s" e using Cons(1,2) by (unfold_locales, auto)
- show ?case
- proof -
- -- {* It can be proved that @{term cntP}-value of @{term th'} does not change
- by the happening of event @{term e}: *}
- have "cntP ((e#t)@s) th' = cntP (t@s) th'"
- proof(rule ccontr) -- {* Proof by contradiction. *}
- -- {* Suppose @{term cntP}-value of @{term th'} is changed by @{term e}: *}
- assume otherwise: "cntP ((e # t) @ s) th' \<noteq> cntP (t @ s) th'"
- -- {* Then the actor of @{term e} must be @{term th'} and @{term e}
- must be a @{term P}-event: *}
- hence "isP e" "actor e = th'" by (auto simp:cntP_diff_inv)
- with vat_es.actor_inv
- -- {* According to @{thm vat_es.actor_inv}, @{term th'} must be running at
- the moment @{term "t@s"}: *}
- have "th' \<in> runing (t@s)" by (cases e, auto)
- -- {* However, an application of @{thm eq_pv_blocked} to induction hypothesis
- shows @{term th'} can not be running at moment @{term "t@s"}: *}
- moreover have "th' \<notin> runing (t@s)"
- using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] .
- -- {* Contradiction is finally derived: *}
- ultimately show False by simp
- qed
- -- {* It can also be proved that @{term cntV}-value of @{term th'} does not change
- by the happening of event @{term e}: *}
- -- {* The proof follows exactly the same pattern as the case for @{term cntP}-value: *}
- moreover have "cntV ((e#t)@s) th' = cntV (t@s) th'"
- proof(rule ccontr) -- {* Proof by contradiction. *}
- assume otherwise: "cntV ((e # t) @ s) th' \<noteq> cntV (t @ s) th'"
- hence "isV e" "actor e = th'" by (auto simp:cntV_diff_inv)
- with vat_es.actor_inv
- have "th' \<in> runing (t@s)" by (cases e, auto)
- moreover have "th' \<notin> runing (t@s)"
- using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] .
- ultimately show False by simp
- qed
- -- {* Finally, it can be shown that the @{term cntP} and @{term cntV}
- value for @{term th'} are still in balance, so @{term th'}
- is still hand-emptied after the execution of event @{term e}: *}
- ultimately show ?thesis using Cons(5) by metis
- qed
-qed (auto simp:eq_pv)
-
-text {*
- By combining @{thm eq_pv_blocked} and @{thm eq_pv_persist},
- it can be derived easily that @{term th'} can not be running in the future:
-*}
-lemma eq_pv_blocked_persist:
- assumes neq_th': "th' \<noteq> th"
- and eq_pv: "cntP s th' = cntV s th'"
- shows "th' \<notin> runing (t@s)"
- using assms
- by (simp add: eq_pv_blocked eq_pv_persist)
-
-text {*
- The following lemma shows the blocking thread @{term th'}
- must hold some resource in the very beginning.
-*}
-lemma runing_cntP_cntV_inv: (* ddd *)
- assumes is_runing: "th' \<in> runing (t@s)"
- and neq_th': "th' \<noteq> th"
- shows "cntP s th' > cntV s th'"
- using assms
-proof -
- -- {* First, it can be shown that the number of @{term P} and
- @{term V} operations can not be equal for thred @{term th'} *}
- have "cntP s th' \<noteq> cntV s th'"
- proof
- -- {* The proof goes by contradiction, suppose otherwise: *}
- assume otherwise: "cntP s th' = cntV s th'"
- -- {* By applying @{thm eq_pv_blocked_persist} to this: *}
- from eq_pv_blocked_persist[OF neq_th' otherwise]
- -- {* we have that @{term th'} can not be running at moment @{term "t@s"}: *}
- have "th' \<notin> runing (t@s)" .
- -- {* This is obvious in contradiction with assumption @{thm is_runing} *}
- thus False using is_runing by simp
- qed
- -- {* However, the number of @{term V} is always less or equal to @{term P}: *}
- moreover have "cntV s th' \<le> cntP s th'" using vat_s.cnp_cnv_cncs by auto
- -- {* Thesis is finally derived by combining the these two results: *}
- ultimately show ?thesis by auto
-qed
-
-
-text {*
- The following lemmas shows the blocking thread @{text th'} must be live
- at the very beginning, i.e. the moment (or state) @{term s}.
-
- The proof is a simple combination of the results above:
-*}
-lemma runing_threads_inv:
- assumes runing': "th' \<in> runing (t@s)"
- and neq_th': "th' \<noteq> th"
- shows "th' \<in> threads s"
-proof(rule ccontr) -- {* Proof by contradiction: *}
- assume otherwise: "th' \<notin> threads s"
- have "th' \<notin> runing (t @ s)"
- proof -
- from vat_s.cnp_cnv_eq[OF otherwise]
- have "cntP s th' = cntV s th'" .
- from eq_pv_blocked_persist[OF neq_th' this]
- show ?thesis .
- qed
- with runing' show False by simp
-qed
-
-text {*
- The following lemma summarizes several foregoing
- lemmas to give an overall picture of the blocking thread @{text "th'"}:
-*}
-lemma runing_inversion: (* ddd, one of the main lemmas to present *)
- 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"
-proof -
- from runing_threads_inv[OF assms]
- show "th' \<in> threads s" .
-next
- from runing_cntP_cntV_inv[OF runing' neq_th]
- show "\<not>detached s th'" using vat_s.detached_eq by simp
-next
- from runing_preced_inversion[OF runing']
- show "cp (t@s) th' = preced th s" .
-qed
-
-section {* The existence of `blocking thread` *}
-
-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, the other main lemma to be presented: *)
- 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.rg_RAG_threads 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
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