updated Correctness, Implementation and PIPBasics so that they work with Isabelle 2014 and 2015
theory Test
imports Precedence_ord Graphs
begin
type_synonym thread = nat -- {* Type for thread identifiers. *}
type_synonym priority = nat -- {* Type for priorities. *}
type_synonym cs = nat -- {* Type for critical sections (or resources). *}
-- {* Schedulling Events *}
datatype event =
Create thread priority
| Exit thread
| P thread cs
| V thread cs
| Set thread priority
type_synonym state = "event list"
fun threads :: "state \<Rightarrow> thread set"
where
"threads [] = {}"
| "threads (Create th prio#s) = {th} \<union> threads s"
| "threads (Exit th # s) = (threads s) - {th}"
| "threads (_#s) = threads s"
fun priority :: "thread \<Rightarrow> state \<Rightarrow> priority"
where
"priority th [] = 0"
| "priority th (Create th' prio#s) = (if th' = th then prio else priority th s)"
| "priority th (Set th' prio#s) = (if th' = th then prio else priority th s)"
| "priority th (_#s) = priority th s"
fun last_set :: "thread \<Rightarrow> state \<Rightarrow> nat"
where
"last_set th [] = 0"
| "last_set th ((Create th' prio)#s) = (if (th = th') then length s else last_set th s)"
| "last_set th ((Set th' prio)#s) = (if (th = th') then length s else last_set th s)"
| "last_set th (_#s) = last_set th s"
definition preced :: "thread \<Rightarrow> state \<Rightarrow> precedence"
where "preced th s \<equiv> Prc (priority th s) (last_set th s)"
abbreviation
"preceds s ths \<equiv> {preced th s | th. th \<in> ths}"
definition
"holds wq th cs \<equiv> th \<in> set (wq cs) \<and> th = hd (wq cs)"
definition
"waits wq th cs \<equiv> th \<in> set (wq cs) \<and> th \<noteq> hd (wq cs)"
--{* Nodes in Resource Graph *}
datatype node =
Th "thread"
| Cs "cs"
definition
"RAG wq \<equiv> {(Th th, Cs cs) | th cs. waits wq th cs} \<union> {(Cs cs, Th th) | cs th. holds wq th cs}"
definition
"dependants wq th \<equiv> {th' . (Th th', Th th) \<in> (RAG wq)^+}"
record schedule_state =
wq_fun :: "cs \<Rightarrow> thread list"
cprec_fun :: "thread \<Rightarrow> precedence"
definition cpreced :: "(cs \<Rightarrow> thread list) \<Rightarrow> state \<Rightarrow> thread \<Rightarrow> precedence"
where
"cpreced wq s th \<equiv> Max ({preced th s} \<union> {preced th' s | th'. th' \<in> dependants wq th})"
abbreviation
"all_unlocked \<equiv> \<lambda>_::cs. ([]::thread list)"
abbreviation
"initial_cprec \<equiv> \<lambda>_::thread. Prc 0 0"
abbreviation
"release qs \<equiv> case qs of
[] => []
| (_ # qs) => SOME q. distinct q \<and> set q = set qs"
lemma [simp]:
"(SOME q. distinct q \<and> q = []) = []"
by auto
lemma [simp]:
"(x \<in> set (SOME q. distinct q \<and> set q = set p)) = (x \<in> set p)"
apply(rule iffI)
apply (metis (mono_tags, lifting) List.finite_set finite_distinct_list some_eq_ex)+
done
abbreviation
"next_to_run ths \<equiv> hd (SOME q::thread list. distinct q \<and> set q = set ths)"
fun schs :: "state \<Rightarrow> schedule_state"
where
"schs [] = (| wq_fun = \<lambda> cs. [], cprec_fun = (\<lambda>_. Prc 0 0) |)"
| "schs (Create th prio # s) =
(let wq = wq_fun (schs s) in
(|wq_fun = wq, cprec_fun = cpreced wq (Create th prio # s)|))"
| "schs (Exit th # s) =
(let wq = wq_fun (schs s) in
(|wq_fun = wq, cprec_fun = cpreced wq (Exit th # s)|))"
| "schs (Set th prio # s) =
(let wq = wq_fun (schs s) in
(|wq_fun = wq, cprec_fun = cpreced wq (Set th prio # s)|))"
| "schs (P th cs # s) =
(let wq = wq_fun (schs s) in
let new_wq = wq(cs := (wq cs @ [th])) in
(|wq_fun = new_wq, cprec_fun = cpreced new_wq (P th cs # s)|))"
| "schs (V th cs # s) =
(let wq = wq_fun (schs s) in
let new_wq = wq(cs := release (wq cs)) in
(|wq_fun = new_wq, cprec_fun = cpreced new_wq (V th cs # s)|))"
definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list"
where "wq s = wq_fun (schs s)"
definition cpreced2 :: "state \<Rightarrow> thread \<Rightarrow> precedence"
where "cpreced2 s \<equiv> cprec_fun (schs s)"
abbreviation
"cpreceds2 s ths \<equiv> {cpreced2 s th | th. th \<in> ths}"
definition
"holds2 s \<equiv> holds (wq_fun (schs s))"
definition
"waits2 s \<equiv> waits (wq_fun (schs s))"
definition
"RAG2 s \<equiv> RAG (wq_fun (schs s))"
definition
"dependants2 s \<equiv> dependants (wq_fun (schs s))"
(* ready -> is a thread that is not waiting for any resource *)
definition readys :: "state \<Rightarrow> thread set"
where "readys s \<equiv> {th . th \<in> threads s \<and> (\<forall> cs. \<not> waits2 s th cs)}"
definition runing :: "state \<Rightarrow> thread set"
where "runing s \<equiv> {th . th \<in> readys s \<and> cpreced2 s th = Max (cpreceds2 s (readys s))}"
(* all resources a thread hols in a state *)
definition holding :: "state \<Rightarrow> thread \<Rightarrow> cs set"
where "holding s th \<equiv> {cs . holds2 s th cs}"
lemma exists_distinct:
obtains ys where "distinct ys" "set ys = set xs"
by (metis List.finite_set finite_distinct_list)
lemma next_to_run_set [simp]:
"wts \<noteq> [] \<Longrightarrow> next_to_run wts \<in> set wts"
apply(rule exists_distinct[of wts])
by (metis (mono_tags, lifting) hd_in_set set_empty some_eq_ex)
lemma holding_RAG:
"holding s th = {cs . (Cs cs, Th th) \<in> RAG2 s}"
unfolding holding_def RAG2_def RAG_def
unfolding holds2_def holds_def waits_def
by auto
inductive step :: "state \<Rightarrow> event \<Rightarrow> bool"
where
step_Create: "\<lbrakk>th \<notin> threads s\<rbrakk> \<Longrightarrow> step s (Create th prio)"
| step_Exit: "\<lbrakk>th \<in> runing s; holding s th = {}\<rbrakk> \<Longrightarrow> step s (Exit th)"
| step_P: "\<lbrakk>th \<in> runing s; (Cs cs, Th th) \<notin> (RAG2 s)^+\<rbrakk> \<Longrightarrow> step s (P th cs)"
| step_V: "\<lbrakk>th \<in> runing s; holds2 s th cs\<rbrakk> \<Longrightarrow> step s (V th cs)"
| step_Set: "\<lbrakk>th \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set th prio)"
(* valid states *)
inductive vt :: "state \<Rightarrow> bool"
where
vt_nil[intro]: "vt []"
| vt_cons[intro]: "\<lbrakk>vt s; step s e\<rbrakk> \<Longrightarrow> vt (e#s)"
lemma runing_ready:
shows "runing s \<subseteq> readys s"
unfolding runing_def readys_def
by auto
lemma readys_threads:
shows "readys s \<subseteq> threads s"
unfolding readys_def
by auto
lemma wq_threads:
assumes vt: "vt s"
and h: "th \<in> set (wq s cs)"
shows "th \<in> threads s"
using assms
apply(induct)
apply(simp add: wq_def)
apply(erule step.cases)
apply(auto simp add: wq_def Let_def holding_def holds2_def holds_def waits2_def runing_def readys_def)
apply(simp add: waits_def)
apply(auto simp add: waits_def split: if_splits)[1]
apply(auto split: if_splits)
apply(simp only: waits_def)
by (metis insert_iff set_simps(2))
lemma Domain_RAG_threads:
assumes vt: "vt s"
and in_dom: "(Th th) \<in> Domain (RAG2 s)"
shows "th \<in> threads s"
proof -
from in_dom obtain n where "(Th th, n) \<in> RAG2 s" by auto
then obtain cs where "n = Cs cs" "(Th th, Cs cs) \<in> RAG2 s"
unfolding RAG2_def RAG_def by auto
then have "th \<in> set (wq s cs)"
unfolding wq_def RAG_def RAG2_def waits_def by auto
with wq_threads [OF vt] show ?thesis .
qed
lemma dependants_threads:
assumes vt: "vt s"
shows "dependants2 s th \<subseteq> threads s"
proof
fix th1
assume "th1 \<in> dependants2 s th"
then have h: "(Th th1, Th th) \<in> (RAG2 s)\<^sup>+"
unfolding dependants2_def dependants_def RAG2_def by simp
then have "Th th1 \<in> Domain ((RAG2 s)\<^sup>+)" unfolding Domain_def by auto
then have "Th th1 \<in> Domain (RAG2 s)" using trancl_domain by simp
then show "th1 \<in> threads s" using vt by (rule_tac Domain_RAG_threads)
qed
lemma finite_threads:
assumes vt: "vt s"
shows "finite (threads s)"
using vt by (induct) (auto elim: step.cases)
section {* Distinctness of @{const wq} *}
lemma wq_distinct_step:
assumes "step s e" "distinct (wq s cs)"
shows "distinct (wq (e # s) cs)"
using assms
unfolding wq_def
apply(erule_tac step.cases)
apply(auto simp add: RAG2_def RAG_def Let_def)[1]
apply(auto simp add: wq_def Let_def RAG2_def RAG_def holds_def runing_def waits2_def waits_def readys_def)
apply(auto split: list.split)
apply(rule someI2)
apply(auto)
done
lemma wq_distinct:
assumes "vt s"
shows "distinct (wq s cs)"
using assms
apply(induct)
apply(simp add: wq_def)
apply(simp add: wq_distinct_step)
done
section {* Single_Valuedness of @{const waits2}, @{const holds2}, @{const RAG2} *}
lemma waits2_unique:
assumes "vt s"
and "waits2 s th cs1"
and "waits2 s th cs2"
shows "cs1 = cs2"
using assms
apply(induct)
apply(simp add: waits2_def waits_def)
apply(erule step.cases)
apply(auto simp add: Let_def waits2_def waits_def holds_def RAG2_def RAG_def
readys_def runing_def split: if_splits)
apply (metis Nil_is_append_conv hd_append2 list.distinct(1) split_list)
apply (metis Nil_is_append_conv hd_append2 list.distinct(1) split_list)
apply (metis distinct.simps(2) distinct_length_2_or_more list.sel(1) wq_def wq_distinct)
by (metis (full_types, hide_lams) distinct.simps(2) distinct_length_2_or_more list.sel(1) wq_def wq_distinct)
lemma single_valued_waits2:
assumes "vt s"
shows "single_valuedP (waits2 s)"
using assms
unfolding single_valued_def
by (metis Collect_splitD fst_eqD sndI waits2_unique)
lemma single_valued_holds2:
assumes "vt s"
shows "single_valuedP (\<lambda>cs th. holds2 s th cs)"
unfolding single_valued_def holds2_def holds_def by simp
lemma single_valued_RAG2:
assumes "vt s"
shows "single_valued (RAG2 s)"
using single_valued_waits2[OF assms] single_valued_holds2[OF assms]
unfolding RAG2_def RAG_def
apply(rule_tac single_valued_union)
unfolding holds2_def[symmetric] waits2_def[symmetric]
apply(rule single_valued_Collect)
apply(simp)
apply(simp add: inj_on_def)
apply(rule single_valued_Collect)
apply(simp)
apply(simp add: inj_on_def)
apply(auto)
done
section {* Properties of @{const RAG2} under events *}
lemma RAG_Set [simp]:
shows "RAG2 (Set th prio # s) = RAG2 s"
unfolding RAG2_def
by (simp add: Let_def)
lemma RAG_Create [simp]:
"RAG2 (Create th prio # s) = RAG2 s"
unfolding RAG2_def
by (simp add: Let_def)
lemma RAG_Exit [simp]:
shows "RAG2 (Exit th # s) = RAG2 s"
unfolding RAG2_def
by (simp add: Let_def)
lemma RAG_P1:
assumes "wq s cs = []"
shows "RAG2 (P th cs # s) \<subseteq> RAG2 s \<union> {(Cs cs, Th th)}"
using assms
unfolding RAG2_def RAG_def wq_def Let_def waits_def holds_def
by (auto simp add: Let_def)
lemma RAG_P2:
assumes "(Cs cs, Th th) \<notin> (RAG2 s)\<^sup>+" "wq s cs \<noteq> []"
shows "RAG2 (P th cs # s) \<subseteq> RAG2 s \<union> {(Th th, Cs cs)}"
using assms
unfolding RAG2_def RAG_def wq_def Let_def waits_def holds_def
by (auto simp add: Let_def)
lemma RAG_V1:
assumes vt: "wq s cs = [th]"
shows "RAG2 (V th cs # s) \<subseteq> RAG2 s - {(Cs cs, Th th)}"
using assms
unfolding RAG2_def RAG_def waits_def holds_def wq_def
by (auto simp add: Let_def)
lemma RAG_V2:
assumes vt:"vt s" "wq s cs = th # wts \<and> wts \<noteq> []"
shows "RAG2 (V th cs # s) \<subseteq>
RAG2 s - {(Cs cs, Th th), (Th (next_to_run wts), Cs cs)} \<union> {(Cs cs, Th (next_to_run wts))}"
unfolding RAG2_def RAG_def waits_def holds_def
using assms wq_distinct[OF vt(1), of"cs"]
by (auto simp add: Let_def wq_def)
section {* Acyclicity of @{const RAG2} *}
lemma acyclic_RAG_step:
assumes vt: "vt s"
and stp: "step s e"
and ac: "acyclic (RAG2 s)"
shows "acyclic (RAG2 (e # s))"
using stp vt ac
proof (induct)
case (step_P th s cs)
have ac: "acyclic (RAG2 s)" by fact
have ds: "(Cs cs, Th th) \<notin> (RAG2 s)\<^sup>+" by fact
{ assume wq_empty: "wq s cs = []" -- "case waiting queue is empty"
then have "(Th th, Cs cs) \<notin> (RAG2 s)\<^sup>+"
proof (rule_tac notI)
assume "(Th th, Cs cs) \<in> (RAG2 s)\<^sup>+"
then obtain x where "(x, Cs cs) \<in> RAG2 s" using tranclD2 by metis
with wq_empty show False by (auto simp: RAG2_def RAG_def wq_def waits_def)
qed
with ac have "acyclic (RAG2 s \<union> {(Cs cs, Th th)})" by simp
then have "acyclic (RAG2 (P th cs # s))" using RAG_P1[OF wq_empty]
by (rule acyclic_subset)
}
moreover
{ assume wq_not_empty: "wq s cs \<noteq> []" -- "case waiting queue is not empty"
from ac ds
have "acyclic (RAG2 s \<union> {(Th th, Cs cs)})" by simp
then have "acyclic (RAG2 (P th cs # s))" using RAG_P2[OF ds wq_not_empty]
by (rule acyclic_subset)
}
ultimately show "acyclic (RAG2 (P th cs # s))" by metis
next
case (step_V th s cs) -- "case for release of a lock"
have vt: "vt s" by fact
have ac: "acyclic (RAG2 s)" by fact
have hd: "holds2 s th cs" by fact
from vt have wq_distinct:"distinct (wq s cs)" by (rule wq_distinct)
from hd have "th \<in> set (wq s cs)" "th = hd (wq s cs)" unfolding holds2_def holds_def wq_def by auto
then obtain wts where eq_wq: "wq s cs = th # wts" by (cases "wq s cs") (auto)
-- "case no thread present in the waiting queue to take over"
{ assume "wts = []"
with eq_wq have "wq s cs = [th]" by simp
then have "RAG2 (V th cs # s) \<subseteq> RAG2 s - {(Cs cs, Th th)}" by (rule RAG_V1)
moreover have "acyclic (RAG2 s - {(Cs cs, Th th)})" using ac by (auto intro: acyclic_subset)
ultimately
have "acyclic (RAG2 (V th cs # s))" by (auto intro: acyclic_subset)
}
moreover
-- "at least one thread present to take over"
{ def nth \<equiv> "next_to_run wts"
assume wq_not_empty: "wts \<noteq> []"
have "waits2 s nth cs"
using eq_wq wq_not_empty wq_distinct
unfolding nth_def waits2_def waits_def wq_def[symmetric] by auto
then have cs_in_RAG: "(Th nth, Cs cs) \<in> RAG2 s"
unfolding RAG2_def RAG_def waits2_def by auto
have "RAG2 (V th cs # s) \<subseteq> RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)} \<union> {(Cs cs, Th nth)}"
unfolding nth_def using vt wq_not_empty eq_wq by (rule_tac RAG_V2) (auto)
moreover
have "acyclic (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)} \<union> {(Cs cs, Th nth)})"
proof -
have "acyclic (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)})" using ac by (auto intro: acyclic_subset)
moreover
have "(Th nth, Cs cs) \<notin> (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)})\<^sup>+"
proof (rule notI)
assume "(Th nth, Cs cs) \<in> (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)})\<^sup>+"
then obtain z where a: "(Th nth, z) \<in> (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)})"
by (metis converse_tranclE)
then have "(Th nth, z) \<in> RAG2 s" by simp
then have "z = Cs cs" using cs_in_RAG single_valued_RAG2[OF vt]
by (simp add: single_valued_def)
then show "False" using a by simp
qed
ultimately
show "acyclic (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)} \<union> {(Cs cs, Th nth) })" by simp
qed
ultimately have "acyclic (RAG2 (V th cs # s))"
by (rule_tac acyclic_subset)
}
ultimately show "acyclic (RAG2 (V th cs # s))" by metis
qed (simp_all)
lemma finite_RAG:
assumes "vt s"
shows "finite (RAG2 s)"
using assms
apply(induct)
apply(simp add: RAG2_def RAG_def waits_def holds_def)
apply(erule step.cases)
apply(auto)
apply(case_tac "wq sa cs = []")
apply(rule finite_subset)
apply(rule RAG_P1)
apply(simp)
apply(simp)
apply(rule finite_subset)
apply(rule RAG_P2)
apply(simp)
apply(simp)
apply(simp)
apply(subgoal_tac "\<exists>wts. wq sa cs = th # wts")
apply(erule exE)
apply(case_tac "wts = []")
apply(rule finite_subset)
apply(rule RAG_V1)
apply(simp)
apply(simp)
apply(rule finite_subset)
apply(rule RAG_V2)
apply(simp)
apply(simp)
apply(simp)
apply(subgoal_tac "th \<in> set (wq sa cs) \<and> th = hd (wq sa cs)")
apply(case_tac "wq sa cs")
apply(auto)[2]
apply(auto simp add: holds2_def holds_def wq_def)
done
lemma dchain_unique:
assumes vt: "vt s"
and th1_d: "(n, Th th1) \<in> (RAG2 s)^+"
and th1_r: "th1 \<in> readys s"
and th2_d: "(n, Th th2) \<in> (RAG2 s)^+"
and th2_r: "th2 \<in> readys s"
shows "th1 = th2"
proof(rule ccontr)
assume neq: "th1 \<noteq> th2"
with single_valued_confluent2 [OF single_valued_RAG2 [OF vt]] th1_d th2_d
have "(Th th1, Th th2) \<in> (RAG2 s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (RAG2 s)\<^sup>+" by auto
moreover
{ assume "(Th th1, Th th2) \<in> (RAG2 s)\<^sup>+"
then obtain n where dd: "(Th th1, n) \<in> RAG2 s" by (metis converse_tranclE)
then obtain cs where eq_n: "n = Cs cs"
unfolding RAG2_def RAG_def by (case_tac n) (auto)
from dd eq_n have "th1 \<notin> readys s"
unfolding RAG2_def RAG_def waits2_def readys_def by (auto)
with th1_r have "False" by auto
}
moreover
{ assume "(Th th2, Th th1) \<in> (RAG2 s)\<^sup>+"
then obtain n where dd: "(Th th2, n) \<in> RAG2 s" by (metis converse_tranclE)
then obtain cs where eq_n: "n = Cs cs"
unfolding RAG2_def RAG_def by (case_tac n) (auto)
from dd eq_n have "th2 \<notin> readys s"
unfolding RAG2_def RAG_def waits2_def readys_def by (auto)
with th2_r have "False" by auto
}
ultimately show "False" by metis
qed
lemma cpreced2_cpreced: "cpreced2 s th = cpreced (wq s) s th"
unfolding cpreced2_def wq_def
apply(induct s rule: schs.induct)
apply(simp add: Let_def cpreced_def dependants_def RAG_def waits_def holds_def preced_def)
apply(simp add: Let_def)
apply(simp add: Let_def)
apply(simp add: Let_def)
apply(subst (2) schs.simps)
apply(simp add: Let_def)
apply(subst (2) schs.simps)
apply(simp add: Let_def)
done
lemma cpreced_Exit:
shows "cpreced2 (Exit th # s) th' = cpreced2 s th'"
by (simp add: cpreced2_cpreced cpreced_def preced_def wq_def Let_def)
lemma readys_Exit:
shows "readys (Exit th # s) = readys s - {th}"
by (auto simp add: readys_def waits2_def Let_def)
lemma readys_Create:
shows "readys (Create th prio # s) \<subseteq> {th} \<union> readys s"
apply (auto simp add: readys_def waits2_def Let_def waits_def)
done
lemma readys_Set:
shows "readys (Set th prio # s) = readys s"
by (auto simp add: readys_def waits2_def Let_def)
lemma readys_P:
shows "readys (P th cs # s) \<subseteq> readys s"
apply(auto simp add: readys_def waits2_def Let_def)
apply(simp add: waits_def)
apply(case_tac "csa = cs")
apply(simp)
apply(drule_tac x="cs" in spec)
apply(simp)
apply (metis hd_append2 in_set_insert insert_Nil list.sel(1))
apply(drule_tac x="csa" in spec)
apply(simp)
done
lemma readys_V:
shows "readys (V th cs # s) \<subseteq> readys s \<union> set (wq s cs)"
apply(auto simp add: readys_def waits2_def waits_def Let_def wq_def)
done
fun the_th :: "node \<Rightarrow> thread"
where "the_th (Th th) = th"
lemma image_Collect2:
"f ` A = {f x | x. x \<in> A}"
apply(auto)
done
lemma Collect_disj_eq2:
"{f x | x. x = y \<or> x \<in> A} = {f y} \<union> {f x | x. x \<in> A}"
by (auto)
lemma last_set_lt:
"th \<in> threads s \<Longrightarrow> last_set th s < length s"
apply(induct rule: threads.induct)
apply(auto)
done
lemma last_set_eq_iff:
assumes "th1 \<in> threads s" "th2 \<in> threads s"
shows "last_set th1 s = last_set th2 s \<longleftrightarrow> th1 = th2"
using assms
apply(induct s rule: threads.induct)
apply(auto split:if_splits dest:last_set_lt)
done
lemma preced_eq_iff:
assumes th_in1: "th1 \<in> threads s"
and th_in2: "th2 \<in> threads s"
shows "preced th1 s = preced th2 s \<longleftrightarrow> th1 = th2"
using assms
by (auto simp add: preced_def last_set_eq_iff)
lemma dm_RAG_threads:
assumes vt: "vt s"
and in_dom: "(Th th) \<in> Domain (RAG2 s)"
shows "th \<in> threads s"
proof -
from in_dom obtain n where a: "(Th th, n) \<in> RAG2 s" by auto
then obtain cs where "n = Cs cs"
unfolding RAG2_def RAG_def
by auto
then have "(Th th, Cs cs) \<in> RAG2 s" using a by simp
hence "th \<in> set (wq s cs)"
unfolding RAG2_def wq_def RAG_def waits_def
by (auto)
then show ?thesis
apply(rule_tac wq_threads)
apply(rule assms)
apply(simp)
done
qed
lemma cpreced_eq_iff:
assumes "th1 \<in> readys s" "th2 \<in> readys s" "vt s"
shows "cpreced2 s th1 = cpreced2 s th2 \<longleftrightarrow> th1 = th2"
proof
def S1\<equiv>"({th1} \<union> dependants (wq s) th1)"
def S2\<equiv>"({th2} \<union> dependants (wq s) th2)"
have fin: "finite ((the_th o fst) ` ((RAG (wq s))\<^sup>+))"
apply(rule)
apply(simp add: finite_trancl)
apply(simp add: wq_def)
apply(rule finite_RAG[simplified RAG2_def])
apply(rule assms)
done
from fin have h: "finite (preceds s S1)" "finite (preceds s S2)"
apply(simp_all add: S2_def S1_def Collect_disj_eq2 image_Collect[symmetric])
apply(rule)
apply(simp add: dependants_def)
apply(rule rev_finite_subset)
apply(assumption)
apply(auto simp add: image_def)[1]
apply(metis fst_conv the_th.simps)
apply(rule)
apply(simp add: dependants_def)
apply(rule rev_finite_subset)
apply(assumption)
apply(auto simp add: image_def)[1]
apply(metis fst_conv the_th.simps)
done
moreover have "S1 \<noteq> {}" "S2 \<noteq> {}" by (simp_all add: S1_def S2_def)
then have "(preceds s S1) \<noteq> {}" "(preceds s S2) \<noteq> {}" by simp_all
ultimately have m: "Max (preceds s S1) \<in> (preceds s S1)" "Max (preceds s S2) \<in> (preceds s S2)"
apply(rule_tac [!] Max_in)
apply(simp_all)
done
assume q: "cpreced2 s th1 = cpreced2 s th2"
then have eq_max: "Max (preceds s S1) = Max (preceds s S2)"
unfolding cpreced2_cpreced cpreced_def
apply(simp only: S1_def S2_def)
apply(simp add: Collect_disj_eq2)
done
obtain th0 where th0_in: "th0 \<in> S1" "th0 \<in> S2" and
eq_f_th1: "preced th0 s = Max (preceds s S1)"
"preced th0 s = Max (preceds s S2)"
using m
apply(clarify)
apply(simp add: eq_max)
apply(subst (asm) (2) preced_eq_iff)
apply(insert assms(2))[1]
apply(simp add: S2_def)
apply(auto)[1]
apply (metis contra_subsetD readys_threads)
apply(simp add: dependants_def)
apply(subgoal_tac "Th tha \<in> Domain ((RAG2 s)^+)")
apply(simp add: trancl_domain)
apply (metis Domain_RAG_threads assms(3))
apply(simp only: RAG2_def wq_def)
apply (metis Domain_iff)
apply(insert assms(1))[1]
apply(simp add: S1_def)
apply(auto)[1]
apply (metis contra_subsetD readys_threads)
apply(simp add: dependants_def)
apply(subgoal_tac "Th th \<in> Domain ((RAG2 s)^+)")
apply(simp add: trancl_domain)
apply (metis Domain_RAG_threads assms(3))
apply(simp only: RAG2_def wq_def)
apply (metis Domain_iff)
apply(simp)
done
then show "th1 = th2"
apply -
apply(insert th0_in assms(1, 2))[1]
apply(simp add: S1_def S2_def)
apply(auto)
--"first case"
prefer 2
apply(subgoal_tac "Th th2 \<in> Domain (RAG2 s)")
apply(subgoal_tac "\<exists>cs. (Th th2, Cs cs) \<in> RAG2 s")
apply(erule exE)
apply(simp add: runing_def RAG2_def RAG_def readys_def waits2_def)[1]
apply(auto simp add: RAG2_def RAG_def)[1]
apply(subgoal_tac "Th th2 \<in> Domain ((RAG2 s)^+)")
apply (metis trancl_domain)
apply(subgoal_tac "(Th th2, Th th1) \<in> (RAG2 s)^+")
apply (metis Domain_iff)
apply(simp add: dependants_def RAG2_def wq_def)
--"second case"
apply(subgoal_tac "Th th1 \<in> Domain (RAG2 s)")
apply(subgoal_tac "\<exists>cs. (Th th1, Cs cs) \<in> RAG2 s")
apply(erule exE)
apply(insert assms(1))[1]
apply(simp add: runing_def RAG2_def RAG_def readys_def waits2_def)[1]
apply(auto simp add: RAG2_def RAG_def)[1]
apply(subgoal_tac "Th th1 \<in> Domain ((RAG2 s)^+)")
apply (metis trancl_domain)
apply(subgoal_tac "(Th th1, Th th2) \<in> (RAG2 s)^+")
apply (metis Domain_iff)
apply(simp add: dependants_def RAG2_def wq_def)
--"third case"
apply(rule dchain_unique)
apply(rule assms(3))
apply(simp add: dependants_def RAG2_def wq_def)
apply(simp)
apply(simp add: dependants_def RAG2_def wq_def)
apply(simp)
done
next
assume "th1 = th2"
then show "cpreced2 s th1 = cpreced2 s th2" by simp
qed
lemma at_most_one_running:
assumes "vt s"
shows "card (runing s) \<le> 1"
proof (rule ccontr)
assume "\<not> card (runing s) \<le> 1"
then have "2 \<le> card (runing s)" by auto
moreover
have "finite (runing s)"
by (metis `\<not> card (runing s) \<le> 1` card_infinite le0)
ultimately obtain th1 th2 where a:
"th1 \<noteq> th2" "th1 \<in> runing s" "th2 \<in> runing s"
"cpreced2 s th1 = cpreced2 s th2"
apply(auto simp add: numerals card_le_Suc_iff runing_def)
apply(blast)
done
then have "th1 = th2"
apply(subst (asm) cpreced_eq_iff)
apply(auto intro: assms a)
apply (metis contra_subsetD runing_ready)+
done
then show "False" using a(1) by auto
qed
(*
obtain th0 where th0_in: "th0 \<in> S1 \<and> th0 \<in> S2"
and eq_f_th0: "preced th0 s = Max ((\<lambda>th. preced th s) ` (S1 \<inter> S2))"
proof -
from fin have h1: "finite ((\<lambda>th. preced th s) ` (S1 \<inter> S2))"
apply(simp only: S1_def S2_def)
apply(rule)
apply(rule)
apply(rule)
apply(simp add: dependants_def)
apply(rule rev_finite_subset)
apply(assumption)
apply(auto simp add: image_def)
apply (metis fst_conv the_th.simps)
done
moreover
have "S1 \<inter> S2 \<noteq> {}" apply (simp add: S1_def S2_def)
apply(auto)
done
then have h2: "((\<lambda>th. preced th s) ` (S1 \<union> S2)) \<noteq> {}" by simp
ultimately have "Max ((\<lambda>th. preced th s) ` (S1 \<union> S2)) \<in> ((\<lambda>th. preced th s) ` (S1 \<union> S2))"
apply(rule Max_in)
done
then show ?thesis using that[intro] apply(auto)
apply(erule_tac preced_unique)
done
qed
*)
thm waits_def waits2_def
end