--- a/PIPBasics.thy Sun Feb 07 21:21:53 2016 +0800
+++ b/PIPBasics.thy Mon Feb 08 10:57:01 2016 +0800
@@ -401,6 +401,274 @@
insert assms V, auto simp:cntV_def)
qed (insert assms, auto simp:cntV_def)
+lemma eq_RAG:
+ "RAG (wq s) = RAG s"
+ by (unfold cs_RAG_def s_RAG_def, auto)
+
+text {*
+ The following three lemmas shows the shape
+ of nodes in @{term tRAG}.
+*}
+lemma tRAG_nodeE:
+ assumes "(n1, n2) \<in> tRAG s"
+ obtains th1 th2 where "n1 = Th th1" "n2 = Th th2"
+ using assms
+ by (auto simp: tRAG_def wRAG_def hRAG_def)
+
+lemma tRAG_ancestorsE:
+ assumes "x \<in> ancestors (tRAG s) u"
+ obtains th where "x = Th th"
+proof -
+ from assms have "(u, x) \<in> (tRAG s)^+"
+ by (unfold ancestors_def, auto)
+ from tranclE[OF this] obtain c where "(c, x) \<in> tRAG s" by auto
+ then obtain th where "x = Th th"
+ by (unfold tRAG_alt_def, auto)
+ from that[OF this] show ?thesis .
+qed
+
+lemma subtree_nodeE:
+ assumes "n \<in> subtree (tRAG s) (Th th)"
+ obtains th1 where "n = Th th1"
+proof -
+ show ?thesis
+ proof(rule subtreeE[OF assms])
+ assume "n = Th th"
+ from that[OF this] show ?thesis .
+ next
+ assume "Th th \<in> ancestors (tRAG s) n"
+ hence "(n, Th th) \<in> (tRAG s)^+" by (auto simp:ancestors_def)
+ hence "\<exists> th1. n = Th th1"
+ proof(induct)
+ case (base y)
+ from tRAG_nodeE[OF this] show ?case by metis
+ next
+ case (step y z)
+ thus ?case by auto
+ qed
+ with that show ?thesis by auto
+ qed
+qed
+
+text {*
+ The following lemmas relate @{term tRAG} with
+ @{term RAG} from different perspectives.
+*}
+
+lemma tRAG_star_RAG: "(tRAG s)^* \<subseteq> (RAG s)^*"
+proof -
+ have "(wRAG s O hRAG s)^* \<subseteq> (RAG s O RAG s)^*"
+ by (rule rtrancl_mono, auto simp:RAG_split)
+ also have "... \<subseteq> ((RAG s)^*)^*"
+ by (rule rtrancl_mono, auto)
+ also have "... = (RAG s)^*" by simp
+ finally show ?thesis by (unfold tRAG_def, simp)
+qed
+
+lemma tRAG_subtree_RAG: "subtree (tRAG s) x \<subseteq> subtree (RAG s) x"
+proof -
+ { fix a
+ assume "a \<in> subtree (tRAG s) x"
+ hence "(a, x) \<in> (tRAG s)^*" by (auto simp:subtree_def)
+ with tRAG_star_RAG
+ have "(a, x) \<in> (RAG s)^*" by auto
+ hence "a \<in> subtree (RAG s) x" by (auto simp:subtree_def)
+ } thus ?thesis by auto
+qed
+
+lemma tRAG_trancl_eq:
+ "{th'. (Th th', Th th) \<in> (tRAG s)^+} =
+ {th'. (Th th', Th th) \<in> (RAG s)^+}"
+ (is "?L = ?R")
+proof -
+ { fix th'
+ assume "th' \<in> ?L"
+ hence "(Th th', Th th) \<in> (tRAG s)^+" by auto
+ from tranclD[OF this]
+ obtain z where h: "(Th th', z) \<in> tRAG s" "(z, Th th) \<in> (tRAG s)\<^sup>*" by auto
+ from tRAG_subtree_RAG and this(2)
+ have "(z, Th th) \<in> (RAG s)^*" by (meson subsetCE tRAG_star_RAG)
+ moreover from h(1) have "(Th th', z) \<in> (RAG s)^+" using tRAG_alt_def by auto
+ ultimately have "th' \<in> ?R" by auto
+ } moreover
+ { fix th'
+ assume "th' \<in> ?R"
+ hence "(Th th', Th th) \<in> (RAG s)^+" by (auto)
+ from plus_rpath[OF this]
+ obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \<noteq> []" by auto
+ hence "(Th th', Th th) \<in> (tRAG s)^+"
+ proof(induct xs arbitrary:th' th rule:length_induct)
+ case (1 xs th' th)
+ then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto)
+ show ?case
+ proof(cases "xs1")
+ case Nil
+ from 1(2)[unfolded Cons1 Nil]
+ have rp: "rpath (RAG s) (Th th') [x1] (Th th)" .
+ hence "(Th th', x1) \<in> (RAG s)"
+ by (cases, auto)
+ then obtain cs where "x1 = Cs cs"
+ by (unfold s_RAG_def, auto)
+ from rpath_nnl_lastE[OF rp[unfolded this]]
+ show ?thesis by auto
+ next
+ case (Cons x2 xs2)
+ from 1(2)[unfolded Cons1[unfolded this]]
+ have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" .
+ from rpath_edges_on[OF this]
+ have eds: "edges_on (Th th' # x1 # x2 # xs2) \<subseteq> RAG s" .
+ have "(Th th', x1) \<in> edges_on (Th th' # x1 # x2 # xs2)"
+ by (simp add: edges_on_unfold)
+ with eds have rg1: "(Th th', x1) \<in> RAG s" by auto
+ then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto)
+ have "(x1, x2) \<in> edges_on (Th th' # x1 # x2 # xs2)"
+ by (simp add: edges_on_unfold)
+ from this eds
+ have rg2: "(x1, x2) \<in> RAG s" by auto
+ from this[unfolded eq_x1]
+ obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto)
+ from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2]
+ have rt1: "(Th th', Th th1) \<in> tRAG s" by (unfold tRAG_alt_def, auto)
+ from rp have "rpath (RAG s) x2 xs2 (Th th)"
+ by (elim rpath_ConsE, simp)
+ from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" .
+ show ?thesis
+ proof(cases "xs2 = []")
+ case True
+ from rpath_nilE[OF rp'[unfolded this]]
+ have "th1 = th" by auto
+ from rt1[unfolded this] show ?thesis by auto
+ next
+ case False
+ from 1(1)[rule_format, OF _ rp' this, unfolded Cons1 Cons]
+ have "(Th th1, Th th) \<in> (tRAG s)\<^sup>+" by simp
+ with rt1 show ?thesis by auto
+ qed
+ qed
+ qed
+ hence "th' \<in> ?L" by auto
+ } ultimately show ?thesis by blast
+qed
+
+lemma tRAG_trancl_eq_Th:
+ "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} =
+ {Th th' | th'. (Th th', Th th) \<in> (RAG s)^+}"
+ using tRAG_trancl_eq by auto
+
+lemma tRAG_Field:
+ "Field (tRAG s) \<subseteq> Field (RAG s)"
+ by (unfold tRAG_alt_def Field_def, auto)
+
+lemma tRAG_mono:
+ assumes "RAG s' \<subseteq> RAG s"
+ shows "tRAG s' \<subseteq> tRAG s"
+ using assms
+ by (unfold tRAG_alt_def, auto)
+
+lemma tRAG_subtree_eq:
+ "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th' \<in> (subtree (RAG s) (Th th))}"
+ (is "?L = ?R")
+proof -
+ { fix n
+ assume h: "n \<in> ?L"
+ hence "n \<in> ?R"
+ by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG)
+ } moreover {
+ fix n
+ assume "n \<in> ?R"
+ then obtain th' where h: "n = Th th'" "(Th th', Th th) \<in> (RAG s)^*"
+ by (auto simp:subtree_def)
+ from rtranclD[OF this(2)]
+ have "n \<in> ?L"
+ proof
+ assume "Th th' \<noteq> Th th \<and> (Th th', Th th) \<in> (RAG s)\<^sup>+"
+ with h have "n \<in> {Th th' | th'. (Th th', Th th) \<in> (RAG s)^+}" by auto
+ thus ?thesis using subtree_def tRAG_trancl_eq by fastforce
+ qed (insert h, auto simp:subtree_def)
+ } ultimately show ?thesis by auto
+qed
+
+lemma threads_set_eq:
+ "the_thread ` (subtree (tRAG s) (Th th)) =
+ {th'. Th th' \<in> (subtree (RAG s) (Th th))}" (is "?L = ?R")
+ by (auto intro:rev_image_eqI simp:tRAG_subtree_eq)
+
+lemma RAG_tRAG_transfer:
+ assumes "RAG s' = RAG s \<union> {(Th th, Cs cs)}"
+ and "(Cs cs, Th th'') \<in> RAG s"
+ shows "tRAG s' = tRAG s \<union> {(Th th, Th th'')}" (is "?L = ?R")
+proof -
+ { fix n1 n2
+ assume "(n1, n2) \<in> ?L"
+ from this[unfolded tRAG_alt_def]
+ obtain th1 th2 cs' where
+ h: "n1 = Th th1" "n2 = Th th2"
+ "(Th th1, Cs cs') \<in> RAG s'"
+ "(Cs cs', Th th2) \<in> RAG s'" by auto
+ from h(4) and assms(1) have cs_in: "(Cs cs', Th th2) \<in> RAG s" by auto
+ from h(3) and assms(1)
+ have "(Th th1, Cs cs') = (Th th, Cs cs) \<or>
+ (Th th1, Cs cs') \<in> RAG s" by auto
+ hence "(n1, n2) \<in> ?R"
+ proof
+ assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)"
+ with assms(1) have "(Th th1, Cs cs) \<in> RAG s'" by auto
+ moreover have "th2 = th''"
+ proof -
+ 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)
+ from held_unique[OF this]
+ show ?thesis by simp
+ qed
+ ultimately show ?thesis using h(1,2) h1 by auto
+ next
+ assume "(Th th1, Cs cs') \<in> RAG s"
+ with cs_in have "(Th th1, Th th2) \<in> tRAG s"
+ by (unfold tRAG_alt_def, auto)
+ from this[folded h(1, 2)] show ?thesis by auto
+ qed
+ } moreover {
+ fix n1 n2
+ assume "(n1, n2) \<in> ?R"
+ hence "(n1, n2) \<in>tRAG s \<or> (n1, n2) = (Th th, Th th'')" by auto
+ hence "(n1, n2) \<in> ?L"
+ proof
+ assume "(n1, n2) \<in> tRAG s"
+ moreover have "... \<subseteq> ?L"
+ proof(rule tRAG_mono)
+ show "RAG s \<subseteq> RAG s'" by (unfold assms(1), auto)
+ qed
+ ultimately show ?thesis by auto
+ next
+ assume eq_n: "(n1, n2) = (Th th, Th th'')"
+ from assms(1, 2) have "(Cs cs, Th th'') \<in> RAG s'" by auto
+ moreover have "(Th th, Cs cs) \<in> RAG s'" using assms(1) by auto
+ ultimately show ?thesis
+ by (unfold eq_n tRAG_alt_def, auto)
+ qed
+ } ultimately show ?thesis by auto
+qed
+
+text {*
+ The following lemmas gives an alternative definition @{term dependants}
+ in terms of @{term tRAG}.
+*}
+
+lemma dependants_alt_def:
+ "dependants s th = {th'. (Th th', Th th) \<in> (tRAG s)^+}"
+ by (metis eq_RAG s_dependants_def tRAG_trancl_eq)
+
+text {*
+ The following lemmas gives another alternative definition @{term dependants}
+ in terms of @{term RAG}.
+*}
+
+lemma dependants_alt_def1:
+ "dependants (s::state) th = {th'. (Th th', Th th) \<in> (RAG s)^+}"
+ using dependants_alt_def tRAG_trancl_eq by auto
+
section {* Locales used to investigate the execution of PIP *}
text {*
@@ -1180,6 +1448,34 @@
using assms
by (metis Field_def UnE dm_RAG_threads rg_RAG_threads)
+text {*
+ As a corollary, this lemma shows that @{term tRAG}
+ is also covered by the @{term threads}-set.
+*}
+lemma subtree_tRAG_thread:
+ assumes "th \<in> threads s"
+ shows "subtree (tRAG s) (Th th) \<subseteq> Th ` threads s" (is "?L \<subseteq> ?R")
+proof -
+ have "?L = {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
+ by (unfold tRAG_subtree_eq, simp)
+ also have "... \<subseteq> ?R"
+ proof
+ fix x
+ assume "x \<in> {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
+ then obtain th' where h: "x = Th th'" "Th th' \<in> subtree (RAG s) (Th th)" by auto
+ from this(2)
+ show "x \<in> ?R"
+ proof(cases rule:subtreeE)
+ case 1
+ thus ?thesis by (simp add: assms h(1))
+ next
+ case 2
+ thus ?thesis by (metis ancestors_Field dm_RAG_threads h(1) image_eqI)
+ qed
+ qed
+ finally show ?thesis .
+qed
+
end
section {* The change of @{term RAG} *}
@@ -2157,7 +2453,7 @@
proof -
from assms(2)
show ?thesis
- by (cases rule:waiting_esE, insert assms, auto) (* ccc *)
+ by (cases rule:waiting_esE, insert assms, auto)
qed
end
@@ -2519,6 +2815,13 @@
section {* RAG is well-founded *}
+text {*
+ In this section, it is proved that both @{term RAG} and
+ its converse @{term "(RAG s)^-1"} are well-founded.
+ The proof is very simple with the help of
+ already proved fact that @{term RAG} is finite.
+*}
+
context valid_trace
begin
@@ -2541,7 +2844,12 @@
end
-section {* RAG forms a forest (or tree) *}
+section {* RAG forms a finite-branching forest (or tree) *}
+
+text {*
+ With all the well-formedness proofs about @{term RAG} in place,
+ it is easy to show.
+*}
context valid_trace
begin
@@ -2569,7 +2877,7 @@
qed
-section {* Derived properties for parts of RAG *}
+section {* Derived properties for sub-graphs of RAG *}
context valid_trace
begin
@@ -2597,6 +2905,11 @@
end
+text {*
+ It can be shown that @{term tRAG} is also a
+ finite-branch relational tree (or forest):
+*}
+
sublocale valid_trace < rtree_s: rtree "tRAG s"
proof(unfold_locales)
from sgv_tRAG show "single_valued (tRAG s)" .
@@ -2636,282 +2949,6 @@
from this[folded tRAG_def] show "fsubtree (tRAG s)" .
qed
-lemma tRAG_nodeE:
- assumes "(n1, n2) \<in> tRAG s"
- obtains th1 th2 where "n1 = Th th1" "n2 = Th th2"
- using assms
- by (auto simp: tRAG_def wRAG_def hRAG_def)
-
-lemma tRAG_ancestorsE:
- assumes "x \<in> ancestors (tRAG s) u"
- obtains th where "x = Th th"
-proof -
- from assms have "(u, x) \<in> (tRAG s)^+"
- by (unfold ancestors_def, auto)
- from tranclE[OF this] obtain c where "(c, x) \<in> tRAG s" by auto
- then obtain th where "x = Th th"
- by (unfold tRAG_alt_def, auto)
- from that[OF this] show ?thesis .
-qed
-
-lemma subtree_nodeE:
- assumes "n \<in> subtree (tRAG s) (Th th)"
- obtains th1 where "n = Th th1"
-proof -
- show ?thesis
- proof(rule subtreeE[OF assms])
- assume "n = Th th"
- from that[OF this] show ?thesis .
- next
- assume "Th th \<in> ancestors (tRAG s) n"
- hence "(n, Th th) \<in> (tRAG s)^+" by (auto simp:ancestors_def)
- hence "\<exists> th1. n = Th th1"
- proof(induct)
- case (base y)
- from tRAG_nodeE[OF this] show ?case by metis
- next
- case (step y z)
- thus ?case by auto
- qed
- with that show ?thesis by auto
- qed
-qed
-
-lemma tRAG_star_RAG: "(tRAG s)^* \<subseteq> (RAG s)^*"
-proof -
- have "(wRAG s O hRAG s)^* \<subseteq> (RAG s O RAG s)^*"
- by (rule rtrancl_mono, auto simp:RAG_split)
- also have "... \<subseteq> ((RAG s)^*)^*"
- by (rule rtrancl_mono, auto)
- also have "... = (RAG s)^*" by simp
- finally show ?thesis by (unfold tRAG_def, simp)
-qed
-
-lemma tRAG_subtree_RAG: "subtree (tRAG s) x \<subseteq> subtree (RAG s) x"
-proof -
- { fix a
- assume "a \<in> subtree (tRAG s) x"
- hence "(a, x) \<in> (tRAG s)^*" by (auto simp:subtree_def)
- with tRAG_star_RAG
- have "(a, x) \<in> (RAG s)^*" by auto
- hence "a \<in> subtree (RAG s) x" by (auto simp:subtree_def)
- } thus ?thesis by auto
-qed
-
-lemma tRAG_trancl_eq:
- "{th'. (Th th', Th th) \<in> (tRAG s)^+} =
- {th'. (Th th', Th th) \<in> (RAG s)^+}"
- (is "?L = ?R")
-proof -
- { fix th'
- assume "th' \<in> ?L"
- hence "(Th th', Th th) \<in> (tRAG s)^+" by auto
- from tranclD[OF this]
- obtain z where h: "(Th th', z) \<in> tRAG s" "(z, Th th) \<in> (tRAG s)\<^sup>*" by auto
- from tRAG_subtree_RAG and this(2)
- have "(z, Th th) \<in> (RAG s)^*" by (meson subsetCE tRAG_star_RAG)
- moreover from h(1) have "(Th th', z) \<in> (RAG s)^+" using tRAG_alt_def by auto
- ultimately have "th' \<in> ?R" by auto
- } moreover
- { fix th'
- assume "th' \<in> ?R"
- hence "(Th th', Th th) \<in> (RAG s)^+" by (auto)
- from plus_rpath[OF this]
- obtain xs where rp: "rpath (RAG s) (Th th') xs (Th th)" "xs \<noteq> []" by auto
- hence "(Th th', Th th) \<in> (tRAG s)^+"
- proof(induct xs arbitrary:th' th rule:length_induct)
- case (1 xs th' th)
- then obtain x1 xs1 where Cons1: "xs = x1#xs1" by (cases xs, auto)
- show ?case
- proof(cases "xs1")
- case Nil
- from 1(2)[unfolded Cons1 Nil]
- have rp: "rpath (RAG s) (Th th') [x1] (Th th)" .
- hence "(Th th', x1) \<in> (RAG s)"
- by (cases, auto)
- then obtain cs where "x1 = Cs cs"
- by (unfold s_RAG_def, auto)
- from rpath_nnl_lastE[OF rp[unfolded this]]
- show ?thesis by auto
- next
- case (Cons x2 xs2)
- from 1(2)[unfolded Cons1[unfolded this]]
- have rp: "rpath (RAG s) (Th th') (x1 # x2 # xs2) (Th th)" .
- from rpath_edges_on[OF this]
- have eds: "edges_on (Th th' # x1 # x2 # xs2) \<subseteq> RAG s" .
- have "(Th th', x1) \<in> edges_on (Th th' # x1 # x2 # xs2)"
- by (simp add: edges_on_unfold)
- with eds have rg1: "(Th th', x1) \<in> RAG s" by auto
- then obtain cs1 where eq_x1: "x1 = Cs cs1" by (unfold s_RAG_def, auto)
- have "(x1, x2) \<in> edges_on (Th th' # x1 # x2 # xs2)"
- by (simp add: edges_on_unfold)
- from this eds
- have rg2: "(x1, x2) \<in> RAG s" by auto
- from this[unfolded eq_x1]
- obtain th1 where eq_x2: "x2 = Th th1" by (unfold s_RAG_def, auto)
- from rg1[unfolded eq_x1] rg2[unfolded eq_x1 eq_x2]
- have rt1: "(Th th', Th th1) \<in> tRAG s" by (unfold tRAG_alt_def, auto)
- from rp have "rpath (RAG s) x2 xs2 (Th th)"
- by (elim rpath_ConsE, simp)
- from this[unfolded eq_x2] have rp': "rpath (RAG s) (Th th1) xs2 (Th th)" .
- show ?thesis
- proof(cases "xs2 = []")
- case True
- from rpath_nilE[OF rp'[unfolded this]]
- have "th1 = th" by auto
- from rt1[unfolded this] show ?thesis by auto
- next
- case False
- from 1(1)[rule_format, OF _ rp' this, unfolded Cons1 Cons]
- have "(Th th1, Th th) \<in> (tRAG s)\<^sup>+" by simp
- with rt1 show ?thesis by auto
- qed
- qed
- qed
- hence "th' \<in> ?L" by auto
- } ultimately show ?thesis by blast
-qed
-
-lemma tRAG_trancl_eq_Th:
- "{Th th' | th'. (Th th', Th th) \<in> (tRAG s)^+} =
- {Th th' | th'. (Th th', Th th) \<in> (RAG s)^+}"
- using tRAG_trancl_eq by auto
-
-
-lemma tRAG_Field:
- "Field (tRAG s) \<subseteq> Field (RAG s)"
- by (unfold tRAG_alt_def Field_def, auto)
-
-lemma tRAG_mono:
- assumes "RAG s' \<subseteq> RAG s"
- shows "tRAG s' \<subseteq> tRAG s"
- using assms
- by (unfold tRAG_alt_def, auto)
-
-lemma tRAG_subtree_eq:
- "(subtree (tRAG s) (Th th)) = {Th th' | th'. Th th' \<in> (subtree (RAG s) (Th th))}"
- (is "?L = ?R")
-proof -
- { fix n
- assume h: "n \<in> ?L"
- hence "n \<in> ?R"
- by (smt mem_Collect_eq subsetCE subtree_def subtree_nodeE tRAG_subtree_RAG)
- } moreover {
- fix n
- assume "n \<in> ?R"
- then obtain th' where h: "n = Th th'" "(Th th', Th th) \<in> (RAG s)^*"
- by (auto simp:subtree_def)
- from rtranclD[OF this(2)]
- have "n \<in> ?L"
- proof
- assume "Th th' \<noteq> Th th \<and> (Th th', Th th) \<in> (RAG s)\<^sup>+"
- with h have "n \<in> {Th th' | th'. (Th th', Th th) \<in> (RAG s)^+}" by auto
- thus ?thesis using subtree_def tRAG_trancl_eq by fastforce
- qed (insert h, auto simp:subtree_def)
- } ultimately show ?thesis by auto
-qed
-
-lemma threads_set_eq:
- "the_thread ` (subtree (tRAG s) (Th th)) =
- {th'. Th th' \<in> (subtree (RAG s) (Th th))}" (is "?L = ?R")
- by (auto intro:rev_image_eqI simp:tRAG_subtree_eq)
-
-context valid_trace
-begin
-
-lemma RAG_tRAG_transfer:
- assumes "RAG s' = RAG s \<union> {(Th th, Cs cs)}"
- and "(Cs cs, Th th'') \<in> RAG s"
- shows "tRAG s' = tRAG s \<union> {(Th th, Th th'')}" (is "?L = ?R")
-proof -
- { fix n1 n2
- assume "(n1, n2) \<in> ?L"
- from this[unfolded tRAG_alt_def]
- obtain th1 th2 cs' where
- h: "n1 = Th th1" "n2 = Th th2"
- "(Th th1, Cs cs') \<in> RAG s'"
- "(Cs cs', Th th2) \<in> RAG s'" by auto
- from h(4) and assms(1) have cs_in: "(Cs cs', Th th2) \<in> RAG s" by auto
- from h(3) and assms(1)
- have "(Th th1, Cs cs') = (Th th, Cs cs) \<or>
- (Th th1, Cs cs') \<in> RAG s" by auto
- hence "(n1, n2) \<in> ?R"
- proof
- assume h1: "(Th th1, Cs cs') = (Th th, Cs cs)"
- hence eq_th1: "th1 = th" by simp
- moreover have "th2 = th''"
- proof -
- from h1 have "cs' = cs" by simp
- from assms(2) cs_in[unfolded this]
- show ?thesis using unique_RAG by auto
- qed
- ultimately show ?thesis using h(1,2) by auto
- next
- assume "(Th th1, Cs cs') \<in> RAG s"
- with cs_in have "(Th th1, Th th2) \<in> tRAG s"
- by (unfold tRAG_alt_def, auto)
- from this[folded h(1, 2)] show ?thesis by auto
- qed
- } moreover {
- fix n1 n2
- assume "(n1, n2) \<in> ?R"
- hence "(n1, n2) \<in>tRAG s \<or> (n1, n2) = (Th th, Th th'')" by auto
- hence "(n1, n2) \<in> ?L"
- proof
- assume "(n1, n2) \<in> tRAG s"
- moreover have "... \<subseteq> ?L"
- proof(rule tRAG_mono)
- show "RAG s \<subseteq> RAG s'" by (unfold assms(1), auto)
- qed
- ultimately show ?thesis by auto
- next
- assume eq_n: "(n1, n2) = (Th th, Th th'')"
- from assms(1, 2) have "(Cs cs, Th th'') \<in> RAG s'" by auto
- moreover have "(Th th, Cs cs) \<in> RAG s'" using assms(1) by auto
- ultimately show ?thesis
- by (unfold eq_n tRAG_alt_def, auto)
- qed
- } ultimately show ?thesis by auto
-qed
-
-lemma subtree_tRAG_thread:
- assumes "th \<in> threads s"
- shows "subtree (tRAG s) (Th th) \<subseteq> Th ` threads s" (is "?L \<subseteq> ?R")
-proof -
- have "?L = {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
- by (unfold tRAG_subtree_eq, simp)
- also have "... \<subseteq> ?R"
- proof
- fix x
- assume "x \<in> {Th th' |th'. Th th' \<in> subtree (RAG s) (Th th)}"
- then obtain th' where h: "x = Th th'" "Th th' \<in> subtree (RAG s) (Th th)" by auto
- from this(2)
- show "x \<in> ?R"
- proof(cases rule:subtreeE)
- case 1
- thus ?thesis by (simp add: assms h(1))
- next
- case 2
- thus ?thesis by (metis ancestors_Field dm_RAG_threads h(1) image_eqI)
- qed
- qed
- finally show ?thesis .
-qed
-
-lemma eq_RAG:
- "RAG (wq s) = RAG s"
- by (unfold cs_RAG_def s_RAG_def, auto)
-
-lemma dependants_alt_def:
- "dependants s th = {th'. (Th th', Th th) \<in> (tRAG s)^+}"
- by (metis eq_RAG s_dependants_def tRAG_trancl_eq)
-
-lemma dependants_alt_def1:
- "dependants (s::state) th = {th'. (Th th', Th th) \<in> (RAG s)^+}"
- using dependants_alt_def tRAG_trancl_eq by auto
-
-end
-
section {* Chain to readys *}
context valid_trace
@@ -2961,7 +2998,7 @@
qed
text {* \noindent
- The following is just an instance of @{text "chain_building"}.
+ The following lemma is proved easily by instantiating @{thm "chain_building"}.
*}
lemma th_chain_to_ready:
assumes th_in: "th \<in> threads s"