--- a/Implementation.thy Fri Jan 29 09:46:07 2016 +0800
+++ b/Implementation.thy Fri Jan 29 10:51:52 2016 +0800
@@ -2,8 +2,8 @@
This file contains lemmas used to guide the recalculation of current precedence
after every system call (or system operation)
*}
-theory Implementation
-imports PIPBasics
+theory ExtGG
+imports CpsG
begin
text {* (* ddd *)
@@ -35,30 +35,7 @@
section {* The @{term Set} operation *}
-text {* (* ddd *)
- The following locale @{text "step_set_cps"} investigates the recalculation
- after the @{text "Set"} operation.
-*}
-locale step_set_cps =
- fixes s' th prio s
- -- {* @{text "s'"} is the system state before the operation *}
- -- {* @{text "s"} is the system state after the operation *}
- defines s_def : "s \<equiv> (Set th prio#s')"
- -- {* @{text "s"} is assumed to be a legitimate state, from which
- the legitimacy of @{text "s"} can be derived. *}
- assumes vt_s: "vt s"
-
-sublocale step_set_cps < vat_s : valid_trace "s"
-proof
- from vt_s show "vt s" .
-qed
-
-sublocale step_set_cps < vat_s' : valid_trace "s'"
-proof
- from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
-qed
-
-context step_set_cps
+context valid_trace_set
begin
text {* (* ddd *)
@@ -67,26 +44,21 @@
of the operation (or event).
*}
+
lemma eq_preced:
assumes "th' \<noteq> th"
- shows "preced th' s = preced th' s'"
+ shows "preced th' (e#s) = preced th' s"
proof -
from assms show ?thesis
- by (unfold s_def, auto simp:preced_def)
+ by (unfold is_set, auto simp:preced_def)
qed
lemma eq_the_preced:
assumes "th' \<noteq> th"
- shows "the_preced s th' = the_preced s' th'"
+ shows "the_preced (e#s) th' = the_preced s th'"
using assms
by (unfold the_preced_def, intro eq_preced, simp)
-text {*
- The following lemma assures that the resetting of priority does not change the RAG.
-*}
-
-lemma eq_dep: "RAG s = RAG s'"
- by (unfold s_def RAG_set_unchanged, auto)
text {* (* ddd *)
Th following lemma @{text "eq_cp_pre"} says that the priority change of @{text "th"}
@@ -97,18 +69,18 @@
*}
lemma eq_cp_pre:
- assumes nd: "Th th \<notin> subtree (RAG s') (Th th')"
- shows "cp s th' = cp s' th'"
+ 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 s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')}) =
- Max (the_preced s' ` {th'a. Th th'a \<in> subtree (RAG s') (Th th')})"
+ 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 eq_dep by simp
+ have "?S1 = ?S2" using RAG_unchanged by simp
-- {* The function values on the base set are equal as well. *}
moreover have "\<forall> e \<in> ?S2. ?f e = ?g e"
proof
@@ -116,7 +88,7 @@
assume "th1 \<in> ?S2"
with nd have "th1 \<noteq> th" by (auto)
from eq_the_preced[OF this]
- show "the_preced s th1 = the_preced s' th1" .
+ show "the_preced (e#s) th1 = the_preced s th1" .
qed
-- {* Therefore, the image of the functions are equal. *}
ultimately have "(?f ` ?S1) = (?g ` ?S2)" by (auto intro!:f_image_eq)
@@ -131,16 +103,9 @@
*}
lemma th_in_no_subtree:
assumes "th' \<noteq> th"
- shows "Th th \<notin> subtree (RAG s') (Th th')"
+ shows "Th th \<notin> subtree (RAG s) (Th th')"
proof -
- have "th \<in> readys s'"
- proof -
- from step_back_step [OF vt_s[unfolded s_def]]
- have "step s' (Set th prio)" .
- hence "th \<in> runing s'" by (cases, simp)
- thus ?thesis by (simp add:readys_def runing_def)
- qed
- from vat_s'.readys_in_no_subtree[OF this assms(1)]
+ from readys_in_no_subtree[OF th_ready_s assms(1)]
show ?thesis by blast
qed
@@ -151,7 +116,7 @@
*}
lemma eq_cp:
assumes "th' \<noteq> th"
- shows "cp s th' = cp s' th'"
+ shows "cp (e#s) th' = cp s th'"
by (rule eq_cp_pre[OF th_in_no_subtree[OF assms]])
end
@@ -162,73 +127,33 @@
The following @{text "step_v_cps"} is the locale for @{text "V"}-operation.
*}
-locale step_v_cps =
- -- {* @{text "th"} is the initiating thread *}
- -- {* @{text "cs"} is the critical resource release by the @{text "V"}-operation *}
- fixes s' th cs s -- {* @{text "s'"} is the state before operation*}
- defines s_def : "s \<equiv> (V th cs#s')" -- {* @{text "s"} is the state after operation*}
- -- {* @{text "s"} is assumed to be valid, which implies the validity of @{text "s'"} *}
- assumes vt_s: "vt s"
-sublocale step_v_cps < vat_s : valid_trace "s"
-proof
- from vt_s show "vt s" .
-qed
-
-sublocale step_v_cps < vat_s' : valid_trace "s'"
-proof
- from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
-qed
-
-context step_v_cps
+context valid_trace_v
begin
-lemma ready_th_s': "th \<in> readys s'"
- using step_back_step[OF vt_s[unfolded s_def]]
- by (cases, simp add:runing_def)
-
-lemma ancestors_th: "ancestors (RAG s') (Th th) = {}"
+lemma ancestors_th: "ancestors (RAG s) (Th th) = {}"
proof -
- from vat_s'.readys_root[OF ready_th_s']
+ from readys_root[OF th_ready_s]
show ?thesis
by (unfold root_def, simp)
qed
-lemma holding_th: "holding s' th cs"
+lemma edge_of_th:
+ "(Cs cs, Th th) \<in> RAG s"
proof -
- from vt_s[unfolded s_def]
- have " PIP s' (V th cs)" by (cases, simp)
- thus ?thesis by (cases, auto)
-qed
-
-lemma edge_of_th:
- "(Cs cs, Th th) \<in> RAG s'"
-proof -
- from holding_th
+ from holding_th_cs_s
show ?thesis
by (unfold s_RAG_def holding_eq, auto)
qed
lemma ancestors_cs:
- "ancestors (RAG s') (Cs cs) = {Th th}"
+ "ancestors (RAG s) (Cs cs) = {Th th}"
proof -
- have "ancestors (RAG s') (Cs cs) = ancestors (RAG s') (Th th) \<union> {Th th}"
- proof(rule vat_s'.rtree_RAG.ancestors_accum)
- from vt_s[unfolded s_def]
- have " PIP s' (V th cs)" by (cases, simp)
- thus "(Cs cs, Th th) \<in> RAG s'"
- proof(cases)
- assume "holding s' th cs"
- from this[unfolded holding_eq]
- show ?thesis by (unfold s_RAG_def, auto)
- qed
- qed
+ 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
-lemma preced_kept: "the_preced s = the_preced s'"
- by (auto simp: s_def the_preced_def preced_def)
-
end
text {*
@@ -236,147 +161,99 @@
which represents the case when there is another thread @{text "th'"}
to take over the critical resource released by the initiating thread @{text "th"}.
*}
-locale step_v_cps_nt = step_v_cps +
- fixes th'
- -- {* @{text "th'"} is assumed to take over @{text "cs"} *}
- assumes nt: "next_th s' th cs th'"
-context step_v_cps_nt
+context valid_trace_v_n
begin
-text {*
- Lemma @{text "RAG_s"} confirms the change of RAG:
- two edges removed and one added, as shown by the following diagram.
-*}
-
-(*
- RAG before the V-operation
- th1 ----|
- |
- th' ----|
- |----> cs -----|
- th2 ----| |
- | |
- th3 ----| |
- |------> th
- th4 ----| |
- | |
- th5 ----| |
- |----> cs'-----|
- th6 ----|
- |
- th7 ----|
-
- RAG after the V-operation
- th1 ----|
- |
- |----> cs ----> th'
- th2 ----|
- |
- th3 ----|
-
- th4 ----|
- |
- th5 ----|
- |----> cs'----> th
- th6 ----|
- |
- th7 ----|
-*)
-
-lemma sub_RAGs': "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s'"
- using next_th_RAG[OF nt] .
+lemma 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 th') = {Th th, Cs cs}"
+ "ancestors (RAG s) (Th taker) = {Th th, Cs cs}"
proof -
- have "ancestors (RAG s') (Th th') = ancestors (RAG s') (Cs cs) \<union> {Cs cs}"
- proof(rule vat_s'.rtree_RAG.ancestors_accum)
- from sub_RAGs' show "(Th th', Cs cs) \<in> RAG s'" by auto
+ 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 s = (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) \<union>
- {(Cs cs, Th th')}"
-proof -
- from step_RAG_v[OF vt_s[unfolded s_def], folded s_def]
- and nt show ?thesis by (auto intro:next_th_unique)
-qed
+ "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, th'}"
- shows "subtree (RAG s) (Th th1) = subtree (RAG s') (Th th1)" (is "_ = ?R")
+ 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 th', Cs cs)})"
- let ?RAG'' = "?RAG' \<union> {(Cs cs, Th th')}"
+ let ?RAG' = "(RAG s - {(Cs cs, Th th), (Th taker, Cs cs)})"
+ let ?RAG'' = "?RAG' \<union> {(Cs cs, Th taker)}"
have "subtree ?RAG' (Th th1) = ?R"
proof(rule subset_del_subtree_outside)
- show "Range {(Cs cs, Th th), (Th th', Cs cs)} \<inter> subtree (RAG s') (Th th1) = {}"
+ show "Range {(Cs cs, Th th), (Th taker, Cs cs)} \<inter> subtree (RAG s) (Th th1) = {}"
proof -
- have "(Th th) \<notin> subtree (RAG s') (Th th1)"
+ have "(Th th) \<notin> subtree (RAG s) (Th th1)"
proof(rule subtree_refute)
- show "Th th1 \<notin> ancestors (RAG s') (Th th)"
+ show "Th th1 \<notin> ancestors (RAG s) (Th th)"
by (unfold ancestors_th, simp)
next
from assms show "Th th1 \<noteq> Th th" by simp
qed
- moreover have "(Cs cs) \<notin> subtree (RAG s') (Th th1)"
+ moreover have "(Cs cs) \<notin> subtree (RAG s) (Th th1)"
proof(rule subtree_refute)
- show "Th th1 \<notin> ancestors (RAG s') (Cs cs)"
+ show "Th th1 \<notin> ancestors (RAG s) (Cs cs)"
by (unfold ancestors_cs, insert assms, auto)
qed simp
- ultimately have "{Th th, Cs cs} \<inter> subtree (RAG s') (Th th1) = {}" by auto
+ ultimately have "{Th th, Cs cs} \<inter> subtree (RAG s) (Th th1) = {}" by auto
thus ?thesis by simp
qed
qed
moreover have "subtree ?RAG'' (Th th1) = subtree ?RAG' (Th th1)"
proof(rule subtree_insert_next)
- show "Th th' \<notin> subtree (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) (Th th1)"
+ show "Th taker \<notin> subtree (RAG s - {(Cs cs, Th th), (Th taker, Cs cs)}) (Th th1)"
proof(rule subtree_refute)
- show "Th th1 \<notin> ancestors (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) (Th th')"
+ show "Th th1 \<notin> ancestors (RAG s - {(Cs cs, Th th), (Th taker, Cs cs)}) (Th taker)"
(is "_ \<notin> ?R")
proof -
- have "?R \<subseteq> ancestors (RAG s') (Th th')" by (rule ancestors_mono, auto)
+ have "?R \<subseteq> ancestors (RAG s) (Th taker)" by (rule ancestors_mono, auto)
moreover have "Th th1 \<notin> ..." using ancestors_th' assms by simp
ultimately show ?thesis by auto
qed
next
- from assms show "Th th1 \<noteq> Th th'" by simp
+ from assms show "Th th1 \<noteq> Th taker" by simp
qed
qed
ultimately show ?thesis by (unfold RAG_s, simp)
qed
lemma cp_kept:
- assumes "th1 \<notin> {th, th'}"
- shows "cp s th1 = cp s' th1"
- by (unfold cp_alt_def preced_kept subtree_kept[OF assms], simp)
+ 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
-locale step_v_cps_nnt = step_v_cps +
- assumes nnt: "\<And> th'. (\<not> next_th s' th cs th')"
-context step_v_cps_nnt
+context valid_trace_v_e
begin
-lemma RAG_s: "RAG s = RAG s' - {(Cs cs, Th th)}"
-proof -
- from nnt and step_RAG_v[OF vt_s[unfolded s_def], folded s_def]
- show ?thesis by auto
-qed
+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 s) (Th th1) = subtree (RAG s') (Th th1)"
+ shows "subtree (RAG (e#s)) (Th th1) = subtree (RAG s) (Th th1)"
proof(unfold RAG_s, rule subset_del_subtree_outside)
- show "Range {(Cs cs, Th th)} \<inter> subtree (RAG s') (Th th1) = {}"
+ show "Range {(Cs cs, Th th)} \<inter> subtree (RAG s) (Th th1) = {}"
proof -
- have "(Th th) \<notin> subtree (RAG s') (Th th1)"
+ have "(Th th) \<notin> subtree (RAG s) (Th th1)"
proof(rule subtree_refute)
- show "Th th1 \<notin> ancestors (RAG s') (Th th)"
+ show "Th th1 \<notin> ancestors (RAG s) (Th th)"
by (unfold ancestors_th, simp)
next
from assms show "Th th1 \<noteq> Th th" by simp
@@ -387,94 +264,72 @@
lemma cp_kept_1:
assumes "th1 \<noteq> th"
- shows "cp s th1 = cp s' th1"
- by (unfold cp_alt_def preced_kept subtree_kept[OF assms], simp)
+ 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}"
+lemma subtree_cs: "subtree (RAG s) (Cs cs) = {Cs cs}"
proof -
{ fix n
- have "(Cs cs) \<notin> ancestors (RAG s') n"
+ have "(Cs cs) \<notin> ancestors (RAG s) n"
proof
- assume "Cs cs \<in> ancestors (RAG s') n"
- 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
+ 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 h[unfolded this] have "(Th th', Cs cs) \<in> RAG s" .
from this[unfolded s_RAG_def]
- have "waiting (wq s') th' cs" by auto
+ have "waiting (wq s) th' cs" by auto
from this[unfolded cs_waiting_def]
- have "1 < length (wq s' cs)"
- by (cases "wq s' cs", auto)
- from holding_next_thI[OF holding_th this]
- obtain th' where "next_th s' th cs th'" by auto
- with nnt show False by auto
+ 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)"
+ assume "n \<in> subtree (RAG s) (Cs cs)"
hence "n = (Cs cs)"
by (elim subtreeE, insert h, auto)
- } moreover have "(Cs cs) \<in> subtree (RAG s') (Cs cs)"
+ } moreover have "(Cs cs) \<in> subtree (RAG s) (Cs cs)"
by (auto simp:subtree_def)
ultimately show ?thesis by auto
qed
lemma subtree_th:
- "subtree (RAG s) (Th th) = subtree (RAG s') (Th th) - {Cs cs}"
-proof(unfold RAG_s, fold subtree_cs, rule vat_s'.rtree_RAG.subtree_del_inside)
+ "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)"
+ show "(Cs cs, Th th) \<in> edges_in (RAG s) (Th th)"
by (unfold edges_in_def, auto simp:subtree_def)
qed
lemma cp_kept_2:
- shows "cp s th = cp s' th"
- by (unfold cp_alt_def subtree_th preced_kept, auto)
+ shows "cp (e#s) th = cp s th"
+ by (unfold cp_alt_def subtree_th the_preced_es, auto)
lemma eq_cp:
- shows "cp s th' = cp s' th'"
+ shows "cp (e#s) th' = cp s th'"
using cp_kept_1 cp_kept_2
by (cases "th' = th", auto)
+
end
-locale step_P_cps =
- fixes s' th cs s
- defines s_def : "s \<equiv> (P th cs#s')"
- assumes vt_s: "vt s"
-
-sublocale step_P_cps < vat_s : valid_trace "s"
-proof
- from vt_s show "vt s" .
-qed
-
section {* The @{term P} operation *}
-sublocale step_P_cps < vat_s' : valid_trace "s'"
-proof
- from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
-qed
-
-context step_P_cps
+context valid_trace_p
begin
-lemma readys_th: "th \<in> readys s'"
-proof -
- from step_back_step [OF vt_s[unfolded s_def]]
- have "PIP s' (P th cs)" .
- hence "th \<in> runing s'" by (cases, simp)
- thus ?thesis by (simp add:readys_def runing_def)
-qed
-
-lemma root_th: "root (RAG s') (Th th)"
- using readys_root[OF readys_th] .
+lemma 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')"
+ shows "Th th \<notin> subtree (RAG s) (Th th')"
proof
- assume "Th th \<in> subtree (RAG s') (Th th')"
+ assume "Th th \<in> subtree (RAG s) (Th th')"
thus False
proof(cases rule:subtreeE)
case 1
@@ -485,162 +340,140 @@
qed
qed
-lemma preced_kept: "the_preced s = the_preced s'"
- by (auto simp: s_def the_preced_def preced_def)
+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
-locale step_P_cps_ne =step_P_cps +
- fixes th'
- assumes ne: "wq s' cs \<noteq> []"
- defines th'_def: "th' \<equiv> hd (wq s' cs)"
-locale step_P_cps_e =step_P_cps +
- assumes ee: "wq s' cs = []"
-
-context step_P_cps_e
+context valid_trace_p_h
begin
-lemma RAG_s: "RAG s = RAG s' \<union> {(Cs cs, Th th)}"
-proof -
- from ee and step_RAG_p[OF vt_s[unfolded s_def], folded s_def]
- show ?thesis by auto
-qed
-
lemma subtree_kept:
assumes "th' \<noteq> th"
- shows "subtree (RAG s) (Th th') = subtree (RAG s') (Th th')"
-proof(unfold RAG_s, rule subtree_insert_next)
+ 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')" .
+ show "Th th \<notin> subtree (RAG s) (Th th')" .
qed
lemma cp_kept:
assumes "th' \<noteq> th"
- shows "cp s th' = cp s' th'"
+ shows "cp (e#s) th' = cp s th'"
proof -
- have "(the_preced s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')}) =
- (the_preced s' ` {th'a. Th th'a \<in> subtree (RAG s') (Th th')})"
+ have "(the_preced (e#s) ` {th'a. Th th'a \<in> subtree (RAG (e#s)) (Th th')}) =
+ (the_preced s ` {th'a. Th th'a \<in> subtree (RAG s) (Th th')})"
by (unfold preced_kept subtree_kept[OF assms], simp)
thus ?thesis by (unfold cp_alt_def, simp)
qed
end
-context step_P_cps_ne
+context valid_trace_p_w
begin
-lemma RAG_s: "RAG s = RAG s' \<union> {(Th th, Cs cs)}"
-proof -
- from step_RAG_p[OF vt_s[unfolded s_def]] and ne
- show ?thesis by (simp add:s_def)
-qed
+interpretation vat_e: valid_trace "e#s"
+ by (unfold_locales, insert vt_e, simp)
-lemma cs_held: "(Cs cs, Th th') \<in> RAG s'"
-proof -
- have "(Cs cs, Th th') \<in> hRAG s'"
- proof -
- from ne
- have " holding s' th' cs"
- by (unfold th'_def holding_eq cs_holding_def, auto)
- thus ?thesis
- by (unfold hRAG_def, auto)
- qed
- thus ?thesis by (unfold RAG_split, auto)
-qed
+lemma 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 s = tRAG s' \<union> {(Th th, Th th')}"
- using RAG_tRAG_transfer[OF RAG_s cs_held] .
+ "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 s) (Th th)"
- shows "cp s th'' = cp s' th''"
+ assumes "Th th'' \<notin> ancestors (tRAG (e#s)) (Th th)"
+ shows "cp (e#s) th'' = cp s th''"
proof -
- have h: "subtree (tRAG s) (Th th'') = subtree (tRAG s') (Th th'')"
+ have h: "subtree (tRAG (e#s)) (Th th'') = subtree (tRAG s) (Th th'')"
proof -
- have "Th th' \<notin> subtree (tRAG s') (Th th'')"
+ have "Th holder \<notin> subtree (tRAG s) (Th th'')"
proof
- assume "Th th' \<in> subtree (tRAG s') (Th th'')"
+ assume "Th holder \<in> subtree (tRAG s) (Th th'')"
thus False
proof(rule subtreeE)
- assume "Th th' = Th th''"
+ assume "Th holder = Th th''"
from assms[unfolded tRAG_s ancestors_def, folded this]
show ?thesis by auto
next
- assume "Th th'' \<in> ancestors (tRAG s') (Th th')"
- moreover have "... \<subseteq> ancestors (tRAG s) (Th th')"
+ 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 s" by (unfold tRAG_s, auto)
+ show "tRAG s \<subseteq> tRAG (e#s)" by (unfold tRAG_s, auto)
qed
- ultimately have "Th th'' \<in> ancestors (tRAG s) (Th th')" by auto
- moreover have "Th th' \<in> ancestors (tRAG s) (Th th)"
+ 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 s) (Th th)"
+ ultimately have "Th th'' \<in> ancestors (tRAG (e#s)) (Th th)"
by (auto simp:ancestors_def)
with assms show ?thesis by auto
qed
qed
from subtree_insert_next[OF this]
- have "subtree (tRAG s' \<union> {(Th th, Th th')}) (Th th'') = subtree (tRAG s') (Th th'')" .
+ have "subtree (tRAG s \<union> {(Th th, Th holder)}) (Th th'') = subtree (tRAG s) (Th th'')" .
from this[folded tRAG_s] show ?thesis .
qed
show ?thesis by (unfold cp_alt_def1 h preced_kept, simp)
qed
lemma cp_gen_update_stop: (* ddd *)
- assumes "u \<in> ancestors (tRAG s) (Th th)"
- and "cp_gen s u = cp_gen s' u"
- and "y \<in> ancestors (tRAG s) u"
- shows "cp_gen s y = cp_gen s' y"
+ 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_s.fsbttRAGs.wf])
+proof(induct rule:wf_induct[OF vat_e.fsbttRAGs.wf])
case (1 x)
show ?case (is "?L = ?R")
proof -
from tRAG_ancestorsE[OF 1(2)]
obtain th2 where eq_x: "x = Th th2" by blast
- from vat_s.cp_gen_rec[OF this]
+ from vat_e.cp_gen_rec[OF this]
have "?L =
- Max ({the_preced s th2} \<union> cp_gen s ` RTree.children (tRAG s) x)" .
+ Max ({the_preced (e#s) th2} \<union> cp_gen (e#s) ` RTree.children (tRAG (e#s)) x)" .
also have "... =
- Max ({the_preced s' th2} \<union> cp_gen s' ` RTree.children (tRAG s') x)"
-
+ Max ({the_preced s th2} \<union> cp_gen s ` RTree.children (tRAG s) x)"
proof -
- from preced_kept have "the_preced s th2 = the_preced s' th2" by simp
- moreover have "cp_gen s ` RTree.children (tRAG s) x =
- cp_gen s' ` RTree.children (tRAG s') x"
+ 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 s) x = RTree.children (tRAG s') x"
+ have "RTree.children (tRAG (e#s)) x = RTree.children (tRAG s) x"
proof(unfold tRAG_s, rule children_union_kept)
- have start: "(Th th, Th th') \<in> tRAG s"
+ have start: "(Th th, Th holder) \<in> tRAG (e#s)"
by (unfold tRAG_s, auto)
note x_u = 1(2)
- show "x \<notin> Range {(Th th, Th th')}"
+ show "x \<notin> Range {(Th th, Th holder)}"
proof
- assume "x \<in> Range {(Th th, Th th')}"
- hence eq_x: "x = Th th'" using RangeE by auto
+ assume "x \<in> Range {(Th th, Th holder)}"
+ hence eq_x: "x = Th holder" using RangeE by auto
show False
- proof(cases rule:vat_s.rtree_s.ancestors_headE[OF assms(1) start])
+ proof(cases rule:vat_e.ancestors_headE[OF assms(1) start])
case 1
- from x_u[folded this, unfolded eq_x] vat_s.acyclic_tRAG
+ from x_u[folded this, unfolded eq_x] vat_e.acyclic_tRAG
show ?thesis by (auto simp:ancestors_def acyclic_def)
next
case 2
with x_u[unfolded eq_x]
- have "(Th th', Th th') \<in> (tRAG s)^+" by (auto simp:ancestors_def)
- with vat_s.acyclic_tRAG show ?thesis by (auto simp:acyclic_def)
+ have "(Th holder, Th holder) \<in> (tRAG (e#s))^+" by (auto simp:ancestors_def)
+ with vat_e.acyclic_tRAG show ?thesis by (auto simp:acyclic_def)
qed
qed
qed
- moreover have "cp_gen s ` RTree.children (tRAG s) x =
- cp_gen s' ` RTree.children (tRAG s) x" (is "?f ` ?A = ?g ` ?A")
+ 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_s.rtree_s.ancestors_childrenE[case_names in_ch out_ch])
+ proof(cases rule:vat_e.rtree_s.ancestors_childrenE[case_names in_ch out_ch])
case in_ch
show ?thesis
proof(cases "a = u")
@@ -648,58 +481,58 @@
from assms(2)[folded this] show ?thesis .
next
case False
- have a_not_in: "a \<notin> ancestors (tRAG s) (Th th)"
+ have a_not_in: "a \<notin> ancestors (tRAG (e#s)) (Th th)"
proof
- assume a_in': "a \<in> ancestors (tRAG s) (Th th)"
+ assume a_in': "a \<in> ancestors (tRAG (e#s)) (Th th)"
have "a = u"
- proof(rule vat_s.rtree_s.ancestors_children_unique)
- from a_in' a_in show "a \<in> ancestors (tRAG s) (Th th) \<inter>
- RTree.children (tRAG s) x" by auto
+ proof(rule vat_e.rtree_s.ancestors_children_unique)
+ from a_in' a_in show "a \<in> ancestors (tRAG (e#s)) (Th th) \<inter>
+ RTree.children (tRAG (e#s)) x" by auto
next
- from assms(1) in_ch show "u \<in> ancestors (tRAG s) (Th th) \<inter>
- RTree.children (tRAG s) x" by auto
+ 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 s th_a = cp s' th_a" .
+ have "cp (e#s) th_a = cp s th_a" .
from this [unfolded cp_gen_def_cond[OF eq_a], folded eq_a]
show ?thesis .
qed
next
case (out_ch z)
- hence h: "z \<in> ancestors (tRAG s) u" "z \<in> RTree.children (tRAG s) x" by auto
+ hence h: "z \<in> ancestors (tRAG (e#s)) u" "z \<in> RTree.children (tRAG (e#s)) x" by auto
show ?thesis
proof(cases "a = z")
case True
- from h(2) have zx_in: "(z, x) \<in> (tRAG s)" by (auto simp:RTree.children_def)
+ from h(2) have zx_in: "(z, x) \<in> (tRAG (e#s))" by (auto simp:RTree.children_def)
from 1(1)[rule_format, OF this h(1)]
- have eq_cp_gen: "cp_gen s z = cp_gen s' z" .
+ have eq_cp_gen: "cp_gen (e#s) z = cp_gen s z" .
with True show ?thesis by metis
next
case False
from a_in obtain th_a where eq_a: "a = Th th_a"
by (auto simp:RTree.children_def tRAG_alt_def)
- have "a \<notin> ancestors (tRAG s) (Th th)"
+ have "a \<notin> ancestors (tRAG (e#s)) (Th th)"
proof
- assume a_in': "a \<in> ancestors (tRAG s) (Th th)"
+ assume a_in': "a \<in> ancestors (tRAG (e#s)) (Th th)"
have "a = z"
- proof(rule vat_s.rtree_s.ancestors_children_unique)
- from assms(1) h(1) have "z \<in> ancestors (tRAG s) (Th th)"
+ proof(rule vat_e.rtree_s.ancestors_children_unique)
+ from assms(1) h(1) have "z \<in> ancestors (tRAG (e#s)) (Th th)"
by (auto simp:ancestors_def)
- with h(2) show " z \<in> ancestors (tRAG s) (Th th) \<inter>
- RTree.children (tRAG s) x" by auto
+ 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 s) (Th th) \<inter> RTree.children (tRAG s) x"
+ show "a \<in> ancestors (tRAG (e#s)) (Th th) \<inter> RTree.children (tRAG (e#s)) x"
by auto
qed
with False show False by auto
qed
from cp_kept[OF this[unfolded eq_a]]
- have "cp s th_a = cp s' th_a" .
+ have "cp (e#s) th_a = cp s th_a" .
from this[unfolded cp_gen_def_cond[OF eq_a], folded eq_a]
show ?thesis .
qed
@@ -710,21 +543,21 @@
ultimately show ?thesis by simp
qed
also have "... = ?R"
- by (fold vat_s'.cp_gen_rec[OF eq_x], simp)
+ by (fold cp_gen_rec[OF eq_x], simp)
finally show ?thesis .
qed
qed
lemma cp_up:
- assumes "(Th th') \<in> ancestors (tRAG s) (Th th)"
- and "cp s th' = cp s' th'"
- and "(Th th'') \<in> ancestors (tRAG s) (Th th')"
- shows "cp s th'' = cp s' th''"
+ 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 s (Th th'') = cp_gen s' (Th th'')"
+ have "cp_gen (e#s) (Th th'') = cp_gen s (Th th'')"
proof(rule cp_gen_update_stop[OF assms(1) _ assms(3)])
from assms(2) cp_gen_def_cond[OF refl[of "Th th'"]]
- show "cp_gen s (Th th') = cp_gen s' (Th th')" by metis
+ show "cp_gen (e#s) (Th th') = cp_gen s (Th th')" by metis
qed
with cp_gen_def_cond[OF refl[of "Th th''"]]
show ?thesis by metis
@@ -734,50 +567,32 @@
section {* The @{term Create} operation *}
-locale step_create_cps =
- fixes s' th prio s
- defines s_def : "s \<equiv> (Create th prio#s')"
- assumes vt_s: "vt s"
-
-sublocale step_create_cps < vat_s: valid_trace "s"
- by (unfold_locales, insert vt_s, simp)
+context valid_trace_create
+begin
-sublocale step_create_cps < vat_s': valid_trace "s'"
- by (unfold_locales, insert step_back_vt[OF vt_s[unfolded s_def]], simp)
-
-context step_create_cps
-begin
+interpretation vat_e: valid_trace "e#s"
+ by (unfold_locales, insert vt_e, simp)
-lemma RAG_kept: "RAG s = RAG s'"
- by (unfold s_def RAG_create_unchanged, auto)
-
-lemma tRAG_kept: "tRAG s = tRAG s'"
- by (unfold tRAG_alt_def RAG_kept, auto)
+lemma 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 s th' = the_preced s' th'"
- by (unfold s_def the_preced_def preced_def, insert assms, auto)
+ 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')"
-proof -
- from vt_s[unfolded s_def]
- have "PIP s' (Create th prio)" by (cases, simp)
- hence "th \<notin> threads s'" by(cases, simp)
- from vat_s'.not_in_thread_isolated[OF this]
- have "Th th \<notin> Field (RAG s')" .
- with tRAG_Field show ?thesis by auto
-qed
+lemma 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 s th' = cp s' th'"
+ shows "cp (e#s) th' = cp s th'"
proof -
- have "(the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th') =
- (the_preced s' \<circ> the_thread) ` subtree (tRAG s') (Th th')"
+ have "(the_preced (e#s) \<circ> the_thread) ` subtree (tRAG (e#s)) (Th th') =
+ (the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th')"
proof(unfold tRAG_kept, rule f_image_eq)
fix a
- assume a_in: "a \<in> subtree (tRAG s') (Th th')"
+ assume a_in: "a \<in> subtree (tRAG s) (Th th')"
then obtain th_a where eq_a: "a = Th th_a"
proof(cases rule:subtreeE)
case 2
@@ -786,9 +601,9 @@
qed auto
have neq_th_a: "th_a \<noteq> th"
proof -
- have "(Th th) \<notin> subtree (tRAG s') (Th th')"
+ have "(Th th) \<notin> subtree (tRAG s) (Th th')"
proof
- assume "Th th \<in> subtree (tRAG s') (Th th')"
+ assume "Th th \<in> subtree (tRAG s) (Th th')"
thus False
proof(cases rule:subtreeE)
case 2
@@ -800,99 +615,72 @@
with a_in[unfolded eq_a] show ?thesis by auto
qed
from preced_kept[OF this]
- show "(the_preced s \<circ> the_thread) a = (the_preced s' \<circ> the_thread) a"
+ show "(the_preced (e#s) \<circ> the_thread) a = (the_preced s \<circ> the_thread) a"
by (unfold eq_a, simp)
qed
thus ?thesis by (unfold cp_alt_def1, simp)
qed
-lemma children_of_th: "RTree.children (tRAG s) (Th th) = {}"
+lemma children_of_th: "RTree.children (tRAG (e#s)) (Th th) = {}"
proof -
{ fix a
- assume "a \<in> RTree.children (tRAG s) (Th th)"
- hence "(a, Th th) \<in> tRAG s" by (auto simp:RTree.children_def)
+ 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 s th = preced th s"
- by (unfold vat_s.cp_rec children_of_th, simp add:the_preced_def)
+lemma eq_cp_th: "cp (e#s) th = preced th (e#s)"
+ by (unfold vat_e.cp_rec children_of_th, simp add:the_preced_def)
end
-locale step_exit_cps =
- fixes s' th prio s
- defines s_def : "s \<equiv> Exit th # s'"
- assumes vt_s: "vt s"
-sublocale step_exit_cps < vat_s: valid_trace "s"
- by (unfold_locales, insert vt_s, simp)
-
-sublocale step_exit_cps < vat_s': valid_trace "s'"
- by (unfold_locales, insert step_back_vt[OF vt_s[unfolded s_def]], simp)
-
-context step_exit_cps
+context valid_trace_exit
begin
lemma preced_kept:
assumes "th' \<noteq> th"
- shows "the_preced s th' = the_preced s' th'"
- by (unfold s_def the_preced_def preced_def, insert assms, auto)
-
-lemma RAG_kept: "RAG s = RAG s'"
- by (unfold s_def RAG_exit_unchanged, auto)
-
-lemma tRAG_kept: "tRAG s = tRAG s'"
- by (unfold tRAG_alt_def RAG_kept, auto)
+ shows "the_preced (e#s) th' = the_preced s th'"
+ using assms
+ by (unfold the_preced_def is_exit preced_def, simp)
-lemma th_ready: "th \<in> readys s'"
-proof -
- from vt_s[unfolded s_def]
- have "PIP s' (Exit th)" by (cases, simp)
- hence h: "th \<in> runing s' \<and> holdents s' th = {}" by (cases, metis)
- thus ?thesis by (unfold runing_def, auto)
-qed
+lemma tRAG_kept: "tRAG (e#s) = tRAG s"
+ by (unfold tRAG_alt_def RAG_unchanged, auto)
-lemma th_holdents: "holdents s' th = {}"
+lemma th_RAG: "Th th \<notin> Field (RAG s)"
proof -
- from vt_s[unfolded s_def]
- have "PIP s' (Exit th)" by (cases, simp)
- thus ?thesis by (cases, metis)
-qed
-
-lemma th_RAG: "Th th \<notin> Field (RAG s')"
-proof -
- have "Th th \<notin> Range (RAG s')"
+ have "Th th \<notin> Range (RAG s)"
proof
- assume "Th th \<in> Range (RAG s')"
- then obtain cs where "holding (wq s') th cs"
+ 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 th_holdents[unfolded holdents_def]
- show False by (unfold eq_holding, auto)
+ with holdents_th_s[unfolded holdents_def]
+ show False by (unfold holding_eq, auto)
qed
- moreover have "Th th \<notin> Domain (RAG s')"
+ moreover have "Th th \<notin> Domain (RAG s)"
proof
- assume "Th th \<in> Domain (RAG s')"
- then obtain cs where "waiting (wq s') th cs"
+ 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 show False by (unfold readys_def eq_waiting, auto)
+ with th_ready_s show False by (unfold readys_def waiting_eq, auto)
qed
ultimately show ?thesis by (auto simp:Field_def)
qed
-lemma th_tRAG: "(Th th) \<notin> Field (tRAG s')"
- using th_RAG tRAG_Field[of s'] by auto
+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 s th' = cp s' th'"
+ shows "cp (e#s) th' = cp s th'"
proof -
- have "(the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th') =
- (the_preced s' \<circ> the_thread) ` subtree (tRAG s') (Th th')"
+ have "(the_preced (e#s) \<circ> the_thread) ` subtree (tRAG (e#s)) (Th th') =
+ (the_preced s \<circ> the_thread) ` subtree (tRAG s) (Th th')"
proof(unfold tRAG_kept, rule f_image_eq)
fix a
- assume a_in: "a \<in> subtree (tRAG s') (Th th')"
+ assume a_in: "a \<in> subtree (tRAG s) (Th th')"
then obtain th_a where eq_a: "a = Th th_a"
proof(cases rule:subtreeE)
case 2
@@ -901,14 +689,14 @@
qed auto
have neq_th_a: "th_a \<noteq> th"
proof -
- from vat_s'.readys_in_no_subtree[OF th_ready 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
+ 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 s \<circ> the_thread) a = (the_preced s' \<circ> the_thread) a"
+ show "(the_preced (e#s) \<circ> the_thread) a = (the_preced s \<circ> the_thread) a"
by (unfold eq_a, simp)
qed
thus ?thesis by (unfold cp_alt_def1, simp)
@@ -918,3 +706,924 @@
end
+=======
+theory ExtGG
+imports PrioG CpsG
+begin
+
+text {*
+ The following two auxiliary lemmas are used to reason about @{term Max}.
+*}
+lemma image_Max_eqI:
+ assumes "finite B"
+ and "b \<in> B"
+ and "\<forall> x \<in> B. f x \<le> f b"
+ shows "Max (f ` B) = f b"
+ using assms
+ using Max_eqI by blast
+
+lemma image_Max_subset:
+ assumes "finite A"
+ and "B \<subseteq> A"
+ and "a \<in> B"
+ and "Max (f ` A) = f a"
+ shows "Max (f ` B) = f a"
+proof(rule image_Max_eqI)
+ show "finite B"
+ using assms(1) assms(2) finite_subset by auto
+next
+ show "a \<in> B" using assms by simp
+next
+ show "\<forall>x\<in>B. f x \<le> f a"
+ by (metis Max_ge assms(1) assms(2) assms(4)
+ finite_imageI image_eqI subsetCE)
+qed
+
+text {*
+ The following locale @{text "highest_gen"} sets the basic context for our
+ investigation: supposing thread @{text th} holds the highest @{term cp}-value
+ in state @{text s}, which means the task for @{text th} is the
+ most urgent. We want to show that
+ @{text th} is treated correctly by PIP, which means
+ @{text th} will not be blocked unreasonably by other less urgent
+ threads.
+*}
+locale highest_gen =
+ fixes s th prio tm
+ assumes vt_s: "vt s"
+ and threads_s: "th \<in> threads s"
+ and highest: "preced th s = Max ((cp s)`threads s)"
+ -- {* The internal structure of @{term th}'s precedence is exposed:*}
+ and preced_th: "preced th s = Prc prio tm"
+
+-- {* @{term s} is a valid trace, so it will inherit all results derived for
+ a valid trace: *}
+sublocale highest_gen < vat_s: valid_trace "s"
+ by (unfold_locales, insert vt_s, simp)
+
+context highest_gen
+begin
+
+text {*
+ @{term tm} is the time when the precedence of @{term th} is set, so
+ @{term tm} must be a valid moment index into @{term s}.
+*}
+lemma lt_tm: "tm < length s"
+ by (insert preced_tm_lt[OF threads_s preced_th], simp)
+
+text {*
+ Since @{term th} holds the highest precedence and @{text "cp"}
+ is the highest precedence of all threads in the sub-tree of
+ @{text "th"} and @{text th} is among these threads,
+ its @{term cp} must equal to its precedence:
+*}
+lemma eq_cp_s_th: "cp s th = preced th s" (is "?L = ?R")
+proof -
+ have "?L \<le> ?R"
+ by (unfold highest, rule Max_ge,
+ auto simp:threads_s finite_threads)
+ moreover have "?R \<le> ?L"
+ by (unfold vat_s.cp_rec, rule Max_ge,
+ auto simp:the_preced_def vat_s.fsbttRAGs.finite_children)
+ ultimately show ?thesis by auto
+qed
+
+(* ccc *)
+lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
+ by (fold max_cp_eq, unfold eq_cp_s_th, insert highest, simp)
+
+lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
+ by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma highest': "cp s th = Max (cp s ` threads s)"
+proof -
+ from highest_cp_preced max_cp_eq[symmetric]
+ show ?thesis by simp
+qed
+
+end
+
+locale extend_highest_gen = highest_gen +
+ fixes t
+ assumes vt_t: "vt (t@s)"
+ and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+ and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+ and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+sublocale extend_highest_gen < vat_t: valid_trace "t@s"
+ by (unfold_locales, insert vt_t, simp)
+
+lemma step_back_vt_app:
+ assumes vt_ts: "vt (t@s)"
+ shows "vt s"
+proof -
+ from vt_ts show ?thesis
+ proof(induct t)
+ case Nil
+ from Nil show ?case by auto
+ next
+ case (Cons e t)
+ assume ih: " vt (t @ s) \<Longrightarrow> vt s"
+ and vt_et: "vt ((e # t) @ s)"
+ show ?case
+ proof(rule ih)
+ show "vt (t @ s)"
+ proof(rule step_back_vt)
+ from vt_et show "vt (e # t @ s)" by simp
+ qed
+ qed
+ qed
+qed
+
+
+locale red_extend_highest_gen = extend_highest_gen +
+ fixes i::nat
+
+sublocale red_extend_highest_gen < red_moment: extend_highest_gen "s" "th" "prio" "tm" "(moment i t)"
+ apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
+ apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp)
+ by (unfold highest_gen_def, auto dest:step_back_vt_app)
+
+
+context extend_highest_gen
+begin
+
+ lemma ind [consumes 0, case_names Nil Cons, induct type]:
+ assumes
+ h0: "R []"
+ and h2: "\<And> e t. \<lbrakk>vt (t@s); step (t@s) e;
+ extend_highest_gen s th prio tm t;
+ extend_highest_gen s th prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+ shows "R t"
+proof -
+ from vt_t extend_highest_gen_axioms show ?thesis
+ proof(induct t)
+ from h0 show "R []" .
+ next
+ case (Cons e t')
+ assume ih: "\<lbrakk>vt (t' @ s); extend_highest_gen s th prio tm t'\<rbrakk> \<Longrightarrow> R t'"
+ and vt_e: "vt ((e # t') @ s)"
+ and et: "extend_highest_gen s th prio tm (e # t')"
+ from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+ from vt_e and step_back_vt have vt_ts: "vt (t'@s)" by auto
+ show ?case
+ proof(rule h2 [OF vt_ts stp _ _ _ ])
+ show "R t'"
+ proof(rule ih)
+ from et show ext': "extend_highest_gen s th prio tm t'"
+ by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+ next
+ from vt_ts show "vt (t' @ s)" .
+ qed
+ next
+ from et show "extend_highest_gen s th prio tm (e # t')" .
+ next
+ from et show ext': "extend_highest_gen s th prio tm t'"
+ by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+ qed
+ qed
+qed
+
+
+lemma th_kept: "th \<in> threads (t @ s) \<and>
+ preced th (t@s) = preced th s" (is "?Q t")
+proof -
+ show ?thesis
+ proof(induct rule:ind)
+ case Nil
+ from threads_s
+ show ?case
+ by auto
+ next
+ case (Cons e t)
+ interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto
+ interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto
+ show ?case
+ proof(cases e)
+ case (Create thread prio)
+ show ?thesis
+ proof -
+ from Cons and Create have "step (t@s) (Create thread prio)" by auto
+ hence "th \<noteq> thread"
+ proof(cases)
+ case thread_create
+ with Cons show ?thesis by auto
+ qed
+ hence "preced th ((e # t) @ s) = preced th (t @ s)"
+ by (unfold Create, auto simp:preced_def)
+ moreover note Cons
+ ultimately show ?thesis
+ by (auto simp:Create)
+ qed
+ next
+ case (Exit thread)
+ from h_e.exit_diff and Exit
+ have neq_th: "thread \<noteq> th" by auto
+ with Cons
+ show ?thesis
+ by (unfold Exit, auto simp:preced_def)
+ next
+ case (P thread cs)
+ with Cons
+ show ?thesis
+ by (auto simp:P preced_def)
+ next
+ case (V thread cs)
+ with Cons
+ show ?thesis
+ by (auto simp:V preced_def)
+ next
+ case (Set thread prio')
+ show ?thesis
+ proof -
+ from h_e.set_diff_low and Set
+ have "th \<noteq> thread" by auto
+ hence "preced th ((e # t) @ s) = preced th (t @ s)"
+ by (unfold Set, auto simp:preced_def)
+ moreover note Cons
+ ultimately show ?thesis
+ by (auto simp:Set)
+ qed
+ qed
+ qed
+qed
+
+text {*
+ According to @{thm th_kept}, thread @{text "th"} has its living status
+ and precedence kept along the way of @{text "t"}. The following lemma
+ shows that this preserved precedence of @{text "th"} remains as the highest
+ along the way of @{text "t"}.
+
+ The proof goes by induction over @{text "t"} using the specialized
+ induction rule @{thm ind}, followed by case analysis of each possible
+ operations of PIP. All cases follow the same pattern rendered by the
+ generalized introduction rule @{thm "image_Max_eqI"}.
+
+ The very essence is to show that precedences, no matter whether they are newly introduced
+ or modified, are always lower than the one held by @{term "th"},
+ which by @{thm th_kept} is preserved along the way.
+*}
+lemma max_kept: "Max (the_preced (t @ s) ` (threads (t@s))) = preced th s"
+proof(induct rule:ind)
+ case Nil
+ from highest_preced_thread
+ show ?case
+ by (unfold the_preced_def, simp)
+next
+ case (Cons e t)
+ interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto
+ interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto
+ show ?case
+ proof(cases e)
+ case (Create thread prio')
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ -- {* The following is the common pattern of each branch of the case analysis. *}
+ -- {* The major part is to show that @{text "th"} holds the highest precedence: *}
+ have "Max (?f ` ?A) = ?f th"
+ proof(rule image_Max_eqI)
+ show "finite ?A" using h_e.finite_threads by auto
+ next
+ show "th \<in> ?A" using h_e.th_kept by auto
+ next
+ show "\<forall>x\<in>?A. ?f x \<le> ?f th"
+ proof
+ fix x
+ assume "x \<in> ?A"
+ hence "x = thread \<or> x \<in> threads (t@s)" by (auto simp:Create)
+ thus "?f x \<le> ?f th"
+ proof
+ assume "x = thread"
+ thus ?thesis
+ apply (simp add:Create the_preced_def preced_def, fold preced_def)
+ using Create h_e.create_low h_t.th_kept lt_tm preced_leI2 preced_th by force
+ next
+ assume h: "x \<in> threads (t @ s)"
+ from Cons(2)[unfolded Create]
+ have "x \<noteq> thread" using h by (cases, auto)
+ hence "?f x = the_preced (t@s) x"
+ by (simp add:Create the_preced_def preced_def)
+ hence "?f x \<le> Max (the_preced (t@s) ` threads (t@s))"
+ by (simp add: h_t.finite_threads h)
+ also have "... = ?f th"
+ by (metis Cons.hyps(5) h_e.th_kept the_preced_def)
+ finally show ?thesis .
+ qed
+ qed
+ qed
+ -- {* The minor part is to show that the precedence of @{text "th"}
+ equals to preserved one, given by the foregoing lemma @{thm th_kept} *}
+ also have "... = ?t" using h_e.th_kept the_preced_def by auto
+ -- {* Then it follows trivially that the precedence preserved
+ for @{term "th"} remains the maximum of all living threads along the way. *}
+ finally show ?thesis .
+ qed
+ next
+ case (Exit thread)
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ have "Max (?f ` ?A) = ?f th"
+ proof(rule image_Max_eqI)
+ show "finite ?A" using h_e.finite_threads by auto
+ next
+ show "th \<in> ?A" using h_e.th_kept by auto
+ next
+ show "\<forall>x\<in>?A. ?f x \<le> ?f th"
+ proof
+ fix x
+ assume "x \<in> ?A"
+ hence "x \<in> threads (t@s)" by (simp add: Exit)
+ hence "?f x \<le> Max (?f ` threads (t@s))"
+ by (simp add: h_t.finite_threads)
+ also have "... \<le> ?f th"
+ apply (simp add:Exit the_preced_def preced_def, fold preced_def)
+ using Cons.hyps(5) h_t.th_kept the_preced_def by auto
+ finally show "?f x \<le> ?f th" .
+ qed
+ qed
+ also have "... = ?t" using h_e.th_kept the_preced_def by auto
+ finally show ?thesis .
+ qed
+ next
+ case (P thread cs)
+ with Cons
+ show ?thesis by (auto simp:preced_def the_preced_def)
+ next
+ case (V thread cs)
+ with Cons
+ show ?thesis by (auto simp:preced_def the_preced_def)
+ next
+ case (Set thread prio')
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ have "Max (?f ` ?A) = ?f th"
+ proof(rule image_Max_eqI)
+ show "finite ?A" using h_e.finite_threads by auto
+ next
+ show "th \<in> ?A" using h_e.th_kept by auto
+ next
+ show "\<forall>x\<in>?A. ?f x \<le> ?f th"
+ proof
+ fix x
+ assume h: "x \<in> ?A"
+ show "?f x \<le> ?f th"
+ proof(cases "x = thread")
+ case True
+ moreover have "the_preced (Set thread prio' # t @ s) thread \<le> the_preced (t @ s) th"
+ proof -
+ have "the_preced (t @ s) th = Prc prio tm"
+ using h_t.th_kept preced_th by (simp add:the_preced_def)
+ moreover have "prio' \<le> prio" using Set h_e.set_diff_low by auto
+ ultimately show ?thesis by (insert lt_tm, auto simp:the_preced_def preced_def)
+ qed
+ ultimately show ?thesis
+ by (unfold Set, simp add:the_preced_def preced_def)
+ next
+ case False
+ then have "?f x = the_preced (t@s) x"
+ by (simp add:the_preced_def preced_def Set)
+ also have "... \<le> Max (the_preced (t@s) ` threads (t@s))"
+ using Set h h_t.finite_threads by auto
+ also have "... = ?f th" by (metis Cons.hyps(5) h_e.th_kept the_preced_def)
+ finally show ?thesis .
+ qed
+ qed
+ qed
+ also have "... = ?t" using h_e.th_kept the_preced_def by auto
+ finally show ?thesis .
+ qed
+ qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max (the_preced (t@s) ` (threads (t@s)))"
+ by (insert th_kept max_kept, auto)
+
+text {*
+ The reason behind the following lemma is that:
+ Since @{term "cp"} is defined as the maximum precedence
+ of those threads contained in the sub-tree of node @{term "Th th"}
+ in @{term "RAG (t@s)"}, and all these threads are living threads, and
+ @{term "th"} is also among them, the maximum precedence of
+ them all must be the one for @{text "th"}.
+*}
+lemma th_cp_max_preced:
+ "cp (t@s) th = Max (the_preced (t@s) ` (threads (t@s)))" (is "?L = ?R")
+proof -
+ let ?f = "the_preced (t@s)"
+ have "?L = ?f th"
+ proof(unfold cp_alt_def, rule image_Max_eqI)
+ show "finite {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+ proof -
+ have "{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)} =
+ the_thread ` {n . n \<in> subtree (RAG (t @ s)) (Th th) \<and>
+ (\<exists> th'. n = Th th')}"
+ by (smt Collect_cong Setcompr_eq_image mem_Collect_eq the_thread.simps)
+ moreover have "finite ..." by (simp add: vat_t.fsbtRAGs.finite_subtree)
+ ultimately show ?thesis by simp
+ qed
+ next
+ show "th \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+ by (auto simp:subtree_def)
+ next
+ show "\<forall>x\<in>{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}.
+ the_preced (t @ s) x \<le> the_preced (t @ s) th"
+ proof
+ fix th'
+ assume "th' \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+ hence "Th th' \<in> subtree (RAG (t @ s)) (Th th)" by auto
+ moreover have "... \<subseteq> Field (RAG (t @ s)) \<union> {Th th}"
+ by (meson subtree_Field)
+ ultimately have "Th th' \<in> ..." by auto
+ hence "th' \<in> threads (t@s)"
+ proof
+ assume "Th th' \<in> {Th th}"
+ thus ?thesis using th_kept by auto
+ next
+ assume "Th th' \<in> Field (RAG (t @ s))"
+ thus ?thesis using vat_t.not_in_thread_isolated by blast
+ qed
+ thus "the_preced (t @ s) th' \<le> the_preced (t @ s) th"
+ by (metis Max_ge finite_imageI finite_threads image_eqI
+ max_kept th_kept the_preced_def)
+ qed
+ qed
+ also have "... = ?R" by (simp add: max_preced the_preced_def)
+ finally show ?thesis .
+qed
+
+lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
+ using max_cp_eq th_cp_max_preced the_preced_def vt_t by presburger
+
+lemma th_cp_preced: "cp (t@s) th = preced th s"
+ by (fold max_kept, unfold th_cp_max_preced, simp)
+
+lemma preced_less:
+ assumes th'_in: "th' \<in> threads s"
+ and neq_th': "th' \<noteq> th"
+ shows "preced th' s < preced th s"
+ using assms
+by (metis Max.coboundedI finite_imageI highest not_le order.trans
+ preced_linorder rev_image_eqI threads_s vat_s.finite_threads
+ vat_s.le_cp)
+
+text {*
+ Counting of the number of @{term "P"} and @{term "V"} operations
+ is the cornerstone of a large number of the following proofs.
+ The reason is that this counting is quite easy to calculate and
+ convenient to use in the reasoning.
+
+ The following lemma shows that the counting controls whether
+ a thread is running or not.
+*}
+
+lemma pv_blocked_pre:
+ assumes th'_in: "th' \<in> threads (t@s)"
+ and neq_th': "th' \<noteq> th"
+ and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+ shows "th' \<notin> runing (t@s)"
+proof
+ assume otherwise: "th' \<in> runing (t@s)"
+ show False
+ proof -
+ have "th' = th"
+ proof(rule preced_unique)
+ show "preced th' (t @ s) = preced th (t @ s)" (is "?L = ?R")
+ proof -
+ have "?L = cp (t@s) th'"
+ by (unfold cp_eq_cpreced cpreced_def count_eq_dependants[OF eq_pv], simp)
+ also have "... = cp (t @ s) th" using otherwise
+ by (metis (mono_tags, lifting) mem_Collect_eq
+ runing_def th_cp_max vat_t.max_cp_readys_threads)
+ also have "... = ?R" by (metis th_cp_preced th_kept)
+ finally show ?thesis .
+ qed
+ qed (auto simp: th'_in th_kept)
+ moreover have "th' \<noteq> th" using neq_th' .
+ ultimately show ?thesis by simp
+ qed
+qed
+
+lemmas pv_blocked = pv_blocked_pre[folded detached_eq]
+
+lemma runing_precond_pre:
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and eq_pv: "cntP s th' = cntV s th'"
+ and neq_th': "th' \<noteq> th"
+ shows "th' \<in> threads (t@s) \<and>
+ cntP (t@s) th' = cntV (t@s) th'"
+proof(induct rule:ind)
+ case (Cons e t)
+ interpret vat_t: extend_highest_gen s th prio tm t using Cons by simp
+ interpret vat_e: extend_highest_gen s th prio tm "(e # t)" using Cons by simp
+ show ?case
+ proof(cases e)
+ case (P thread cs)
+ show ?thesis
+ proof -
+ have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ proof -
+ have "thread \<noteq> th'"
+ proof -
+ have "step (t@s) (P thread cs)" using Cons P by auto
+ thus ?thesis
+ proof(cases)
+ assume "thread \<in> runing (t@s)"
+ moreover have "th' \<notin> runing (t@s)" using Cons(5)
+ by (metis neq_th' vat_t.pv_blocked_pre)
+ ultimately show ?thesis by auto
+ qed
+ qed with Cons show ?thesis
+ by (unfold P, simp add:cntP_def cntV_def count_def)
+ qed
+ moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold P, simp)
+ ultimately show ?thesis by auto
+ qed
+ next
+ case (V thread cs)
+ show ?thesis
+ proof -
+ have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ proof -
+ have "thread \<noteq> th'"
+ proof -
+ have "step (t@s) (V thread cs)" using Cons V by auto
+ thus ?thesis
+ proof(cases)
+ assume "thread \<in> runing (t@s)"
+ moreover have "th' \<notin> runing (t@s)" using Cons(5)
+ by (metis neq_th' vat_t.pv_blocked_pre)
+ ultimately show ?thesis by auto
+ qed
+ qed with Cons show ?thesis
+ by (unfold V, simp add:cntP_def cntV_def count_def)
+ qed
+ moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold V, simp)
+ ultimately show ?thesis by auto
+ qed
+ next
+ case (Create thread prio')
+ show ?thesis
+ proof -
+ have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ proof -
+ have "thread \<noteq> th'"
+ proof -
+ have "step (t@s) (Create thread prio')" using Cons Create by auto
+ thus ?thesis using Cons(5) by (cases, auto)
+ qed with Cons show ?thesis
+ by (unfold Create, simp add:cntP_def cntV_def count_def)
+ qed
+ moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold Create, simp)
+ ultimately show ?thesis by auto
+ qed
+ next
+ case (Exit thread)
+ show ?thesis
+ proof -
+ have neq_thread: "thread \<noteq> th'"
+ proof -
+ have "step (t@s) (Exit thread)" using Cons Exit by auto
+ thus ?thesis apply (cases) using Cons(5)
+ by (metis neq_th' vat_t.pv_blocked_pre)
+ qed
+ hence "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" using Cons
+ by (unfold Exit, simp add:cntP_def cntV_def count_def)
+ moreover have "th' \<in> threads ((e # t) @ s)" using Cons neq_thread
+ by (unfold Exit, simp)
+ ultimately show ?thesis by auto
+ qed
+ next
+ case (Set thread prio')
+ with Cons
+ show ?thesis
+ by (auto simp:cntP_def cntV_def count_def)
+ qed
+next
+ case Nil
+ with assms
+ show ?case by auto
+qed
+
+text {* Changing counting balance to detachedness *}
+lemmas runing_precond_pre_dtc = runing_precond_pre
+ [folded vat_t.detached_eq vat_s.detached_eq]
+
+lemma runing_precond:
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and neq_th': "th' \<noteq> th"
+ and is_runing: "th' \<in> runing (t@s)"
+ shows "cntP s th' > cntV s th'"
+ using assms
+proof -
+ have "cntP s th' \<noteq> cntV s th'"
+ by (metis is_runing neq_th' pv_blocked_pre runing_precond_pre th'_in)
+ moreover have "cntV s th' \<le> cntP s th'" using vat_s.cnp_cnv_cncs by auto
+ ultimately show ?thesis by auto
+qed
+
+lemma moment_blocked_pre:
+ assumes neq_th': "th' \<noteq> th"
+ and th'_in: "th' \<in> threads ((moment i t)@s)"
+ and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+ shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
+ th' \<in> threads ((moment (i+j) t)@s)"
+proof -
+ interpret h_i: red_extend_highest_gen _ _ _ _ _ i
+ by (unfold_locales)
+ interpret h_j: red_extend_highest_gen _ _ _ _ _ "i+j"
+ by (unfold_locales)
+ interpret h: extend_highest_gen "((moment i t)@s)" th prio tm "moment j (restm i t)"
+ proof(unfold_locales)
+ show "vt (moment i t @ s)" by (metis h_i.vt_t)
+ next
+ show "th \<in> threads (moment i t @ s)" by (metis h_i.th_kept)
+ next
+ show "preced th (moment i t @ s) =
+ Max (cp (moment i t @ s) ` threads (moment i t @ s))"
+ by (metis h_i.th_cp_max h_i.th_cp_preced h_i.th_kept)
+ next
+ show "preced th (moment i t @ s) = Prc prio tm" by (metis h_i.th_kept preced_th)
+ next
+ show "vt (moment j (restm i t) @ moment i t @ s)"
+ using moment_plus_split by (metis add.commute append_assoc h_j.vt_t)
+ next
+ fix th' prio'
+ assume "Create th' prio' \<in> set (moment j (restm i t))"
+ thus "prio' \<le> prio" using assms
+ by (metis Un_iff add.commute h_j.create_low moment_plus_split set_append)
+ next
+ fix th' prio'
+ assume "Set th' prio' \<in> set (moment j (restm i t))"
+ thus "th' \<noteq> th \<and> prio' \<le> prio"
+ by (metis Un_iff add.commute h_j.set_diff_low moment_plus_split set_append)
+ next
+ fix th'
+ assume "Exit th' \<in> set (moment j (restm i t))"
+ thus "th' \<noteq> th"
+ by (metis Un_iff add.commute h_j.exit_diff moment_plus_split set_append)
+ qed
+ show ?thesis
+ by (metis add.commute append_assoc eq_pv h.runing_precond_pre
+ moment_plus_split neq_th' th'_in)
+qed
+
+lemma moment_blocked_eqpv:
+ assumes neq_th': "th' \<noteq> th"
+ and th'_in: "th' \<in> threads ((moment i t)@s)"
+ and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+ and le_ij: "i \<le> j"
+ shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+ th' \<in> threads ((moment j t)@s) \<and>
+ th' \<notin> runing ((moment j t)@s)"
+proof -
+ from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+ have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+ and h2: "th' \<in> threads ((moment j t)@s)" by auto
+ moreover have "th' \<notin> runing ((moment j t)@s)"
+ proof -
+ interpret h: red_extend_highest_gen _ _ _ _ _ j by (unfold_locales)
+ show ?thesis
+ using h.pv_blocked_pre h1 h2 neq_th' by auto
+ qed
+ ultimately show ?thesis by auto
+qed
+
+(* The foregoing two lemmas are preparation for this one, but
+ in long run can be combined. Maybe I am wrong.
+*)
+lemma moment_blocked:
+ assumes neq_th': "th' \<noteq> th"
+ and th'_in: "th' \<in> threads ((moment i t)@s)"
+ and dtc: "detached (moment i t @ s) th'"
+ and le_ij: "i \<le> j"
+ shows "detached (moment j t @ s) th' \<and>
+ th' \<in> threads ((moment j t)@s) \<and>
+ th' \<notin> runing ((moment j t)@s)"
+proof -
+ interpret h_i: red_extend_highest_gen _ _ _ _ _ i by (unfold_locales)
+ interpret h_j: red_extend_highest_gen _ _ _ _ _ j by (unfold_locales)
+ have cnt_i: "cntP (moment i t @ s) th' = cntV (moment i t @ s) th'"
+ by (metis dtc h_i.detached_elim)
+ from moment_blocked_eqpv[OF neq_th' th'_in cnt_i le_ij]
+ show ?thesis by (metis h_j.detached_intro)
+qed
+
+lemma runing_preced_inversion:
+ assumes runing': "th' \<in> runing (t@s)"
+ shows "cp (t@s) th' = preced th s" (is "?L = ?R")
+proof -
+ have "?L = Max (cp (t @ s) ` readys (t @ s))" using assms
+ by (unfold runing_def, auto)
+ also have "\<dots> = ?R"
+ by (metis th_cp_max th_cp_preced vat_t.max_cp_readys_threads)
+ finally show ?thesis .
+qed
+
+text {*
+ The situation when @{term "th"} is blocked is analyzed by the following lemmas.
+*}
+
+text {*
+ The following lemmas shows the running thread @{text "th'"}, if it is different from
+ @{term th}, must be live at the very beginning. By the term {\em the very beginning},
+ we mean the moment where the formal investigation starts, i.e. the moment (or state)
+ @{term s}.
+*}
+
+lemma runing_inversion_0:
+ assumes neq_th': "th' \<noteq> th"
+ and runing': "th' \<in> runing (t@s)"
+ shows "th' \<in> threads s"
+proof -
+ -- {* The proof is by contradiction: *}
+ { assume otherwise: "\<not> ?thesis"
+ have "th' \<notin> runing (t @ s)"
+ proof -
+ -- {* Since @{term "th'"} is running at time @{term "t@s"}, so it exists that time. *}
+ have th'_in: "th' \<in> threads (t@s)" using runing' by (simp add:runing_def readys_def)
+ -- {* However, @{text "th'"} does not exist at very beginning. *}
+ have th'_notin: "th' \<notin> threads (moment 0 t @ s)" using otherwise
+ by (metis append.simps(1) moment_zero)
+ -- {* Therefore, there must be a moment during @{text "t"}, when
+ @{text "th'"} came into being. *}
+ -- {* Let us suppose the moment being @{text "i"}: *}
+ from p_split_gen[OF th'_in th'_notin]
+ obtain i where lt_its: "i < length t"
+ and le_i: "0 \<le> i"
+ and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
+ and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by (auto)
+ interpret h_i: red_extend_highest_gen _ _ _ _ _ i by (unfold_locales)
+ interpret h_i': red_extend_highest_gen _ _ _ _ _ "(Suc i)" by (unfold_locales)
+ from lt_its have "Suc i \<le> length t" by auto
+ -- {* Let us also suppose the event which makes this change is @{text e}: *}
+ from moment_head[OF this] obtain e where
+ eq_me: "moment (Suc i) t = e # moment i t" by blast
+ hence "vt (e # (moment i t @ s))" by (metis append_Cons h_i'.vt_t)
+ hence "PIP (moment i t @ s) e" by (cases, simp)
+ -- {* It can be derived that this event @{text "e"}, which
+ gives birth to @{term "th'"} must be a @{term "Create"}: *}
+ from create_pre[OF this, of th']
+ obtain prio where eq_e: "e = Create th' prio"
+ by (metis append_Cons eq_me lessI post pre)
+ have h1: "th' \<in> threads (moment (Suc i) t @ s)" using post by auto
+ have h2: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+ proof -
+ have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+ by (metis h_i.cnp_cnv_eq pre)
+ thus ?thesis by (simp add:eq_me eq_e cntP_def cntV_def count_def)
+ qed
+ show ?thesis
+ using moment_blocked_eqpv [OF neq_th' h1 h2, of "length t"] lt_its moment_ge
+ by auto
+ qed
+ with `th' \<in> runing (t@s)`
+ have False by simp
+ } thus ?thesis by auto
+qed
+
+text {*
+ The second lemma says, if the running thread @{text th'} is different from
+ @{term th}, then this @{text th'} must in the possession of some resources
+ at the very beginning.
+
+ To ease the reasoning of resource possession of one particular thread,
+ we used two auxiliary functions @{term cntV} and @{term cntP},
+ which are the counters of @{term P}-operations and
+ @{term V}-operations respectively.
+ If the number of @{term V}-operation is less than the number of
+ @{term "P"}-operations, the thread must have some unreleased resource.
+*}
+
+lemma runing_inversion_1: (* ddd *)
+ assumes neq_th': "th' \<noteq> th"
+ and runing': "th' \<in> runing (t@s)"
+ -- {* thread @{term "th'"} is a live on in state @{term "s"} and
+ it has some unreleased resource. *}
+ shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof -
+ -- {* The proof is a simple composition of @{thm runing_inversion_0} and
+ @{thm runing_precond}: *}
+ -- {* By applying @{thm runing_inversion_0} to assumptions,
+ it can be shown that @{term th'} is live in state @{term s}: *}
+ have "th' \<in> threads s" using runing_inversion_0[OF assms(1,2)] .
+ -- {* Then the thesis is derived easily by applying @{thm runing_precond}: *}
+ with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+qed
+
+text {*
+ The following lemma is just a rephrasing of @{thm runing_inversion_1}:
+*}
+lemma runing_inversion_2:
+ assumes runing': "th' \<in> runing (t@s)"
+ shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
+proof -
+ from runing_inversion_1[OF _ runing']
+ show ?thesis by auto
+qed
+
+lemma runing_inversion_3:
+ assumes runing': "th' \<in> runing (t@s)"
+ and neq_th: "th' \<noteq> th"
+ shows "th' \<in> threads s \<and> (cntV s th' < cntP s th' \<and> cp (t@s) th' = preced th s)"
+ by (metis neq_th runing' runing_inversion_2 runing_preced_inversion)
+
+lemma runing_inversion_4:
+ assumes runing': "th' \<in> runing (t@s)"
+ and neq_th: "th' \<noteq> th"
+ shows "th' \<in> threads s"
+ and "\<not>detached s th'"
+ and "cp (t@s) th' = preced th s"
+ apply (metis neq_th runing' runing_inversion_2)
+ apply (metis neq_th pv_blocked runing' runing_inversion_2 runing_precond_pre_dtc)
+ by (metis neq_th runing' runing_inversion_3)
+
+
+text {*
+ Suppose @{term th} is not running, it is first shown that
+ there is a path in RAG leading from node @{term th} to another thread @{text "th'"}
+ in the @{term readys}-set (So @{text "th'"} is an ancestor of @{term th}}).
+
+ Now, since @{term readys}-set is non-empty, there must be
+ one in it which holds the highest @{term cp}-value, which, by definition,
+ is the @{term runing}-thread. However, we are going to show more: this running thread
+ is exactly @{term "th'"}.
+ *}
+lemma th_blockedE: (* ddd *)
+ assumes "th \<notin> runing (t@s)"
+ obtains th' where "Th th' \<in> ancestors (RAG (t @ s)) (Th th)"
+ "th' \<in> runing (t@s)"
+proof -
+ -- {* According to @{thm vat_t.th_chain_to_ready}, either
+ @{term "th"} is in @{term "readys"} or there is path leading from it to
+ one thread in @{term "readys"}. *}
+ have "th \<in> readys (t @ s) \<or> (\<exists>th'. th' \<in> readys (t @ s) \<and> (Th th, Th th') \<in> (RAG (t @ s))\<^sup>+)"
+ using th_kept vat_t.th_chain_to_ready by auto
+ -- {* However, @{term th} can not be in @{term readys}, because otherwise, since
+ @{term th} holds the highest @{term cp}-value, it must be @{term "runing"}. *}
+ moreover have "th \<notin> readys (t@s)"
+ using assms runing_def th_cp_max vat_t.max_cp_readys_threads by auto
+ -- {* So, there must be a path from @{term th} to another thread @{text "th'"} in
+ term @{term readys}: *}
+ ultimately obtain th' where th'_in: "th' \<in> readys (t@s)"
+ and dp: "(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+" by auto
+ -- {* We are going to show that this @{term th'} is running. *}
+ have "th' \<in> runing (t@s)"
+ proof -
+ -- {* We only need to show that this @{term th'} holds the highest @{term cp}-value: *}
+ have "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" (is "?L = ?R")
+ proof -
+ have "?L = Max ((the_preced (t @ s) \<circ> the_thread) ` subtree (tRAG (t @ s)) (Th th'))"
+ by (unfold cp_alt_def1, simp)
+ also have "... = (the_preced (t @ s) \<circ> the_thread) (Th th)"
+ proof(rule image_Max_subset)
+ show "finite (Th ` (threads (t@s)))" by (simp add: vat_t.finite_threads)
+ next
+ show "subtree (tRAG (t @ s)) (Th th') \<subseteq> Th ` threads (t @ s)"
+ by (metis Range.intros dp trancl_range vat_t.range_in vat_t.subtree_tRAG_thread)
+ next
+ show "Th th \<in> subtree (tRAG (t @ s)) (Th th')" using dp
+ by (unfold tRAG_subtree_eq, auto simp:subtree_def)
+ next
+ show "Max ((the_preced (t @ s) \<circ> the_thread) ` Th ` threads (t @ s)) =
+ (the_preced (t @ s) \<circ> the_thread) (Th th)" (is "Max ?L = _")
+ proof -
+ have "?L = the_preced (t @ s) ` threads (t @ s)"
+ by (unfold image_comp, rule image_cong, auto)
+ thus ?thesis using max_preced the_preced_def by auto
+ qed
+ qed
+ also have "... = ?R"
+ using th_cp_max th_cp_preced th_kept
+ the_preced_def vat_t.max_cp_readys_threads by auto
+ finally show ?thesis .
+ qed
+ -- {* Now, since @{term th'} holds the highest @{term cp}
+ and we have already show it is in @{term readys},
+ it is @{term runing} by definition. *}
+ with `th' \<in> readys (t@s)` show ?thesis by (simp add: runing_def)
+ qed
+ -- {* It is easy to show @{term th'} is an ancestor of @{term th}: *}
+ moreover have "Th th' \<in> ancestors (RAG (t @ s)) (Th th)"
+ using `(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+` by (auto simp:ancestors_def)
+ ultimately show ?thesis using that by metis
+qed
+
+text {*
+ Now it is easy to see there is always a thread to run by case analysis
+ on whether thread @{term th} is running: if the answer is Yes, the
+ the running thread is obviously @{term th} itself; otherwise, the running
+ thread is the @{text th'} given by lemma @{thm th_blockedE}.
+*}
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)")
+ case True thus ?thesis by auto
+next
+ case False
+ thus ?thesis using th_blockedE by auto
+qed
+
+end
+end
+