--- 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
-