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(* Title: Pnet.thy
License: BSD 2-Clause. See LICENSE.
Author: Timothy Bourke
*)
section "Lemmas for partial networks"
theory Pnet
imports AWN_SOS Invariants
begin
text \<open>
These lemmas mostly concern the preservation of node structure by @{term pnet_sos} transitions.
\<close>
lemma pnet_maintains_dom:
assumes "(s, a, s') \<in> trans (pnet np p)"
shows "net_ips s = net_ips s'"
using assms proof (induction p arbitrary: s a s')
fix i R \<sigma> s a s'
assume "(s, a, s') \<in> trans (pnet np \<langle>i; R\<rangle>)"
hence "(s, a, s') \<in> node_sos (trans (np i))" ..
thus "net_ips s = net_ips s'"
by (rule node_sos.cases) simp_all
next
fix p1 p2 s a s'
assume "\<And>s a s'. (s, a, s') \<in> trans (pnet np p1) \<Longrightarrow> net_ips s = net_ips s'"
and "\<And>s a s'. (s, a, s') \<in> trans (pnet np p2) \<Longrightarrow> net_ips s = net_ips s'"
and "(s, a, s') \<in> trans (pnet np (p1 \<parallel> p2))"
thus "net_ips s = net_ips s'"
by simp (erule pnet_sos.cases, simp_all)
qed
lemma pnet_net_ips_net_tree_ips [elim]:
assumes "s \<in> reachable (pnet np p) I"
shows "net_ips s = net_tree_ips p"
using assms proof induction
fix s
assume "s \<in> init (pnet np p)"
thus "net_ips s = net_tree_ips p"
proof (induction p arbitrary: s)
fix i R s
assume "s \<in> init (pnet np \<langle>i; R\<rangle>)"
then obtain ns where "s = NodeS i ns R" ..
thus "net_ips s = net_tree_ips \<langle>i; R\<rangle>"
by simp
next
fix p1 p2 s
assume IH1: "\<And>s. s \<in> init (pnet np p1) \<Longrightarrow> net_ips s = net_tree_ips p1"
and IH2: "\<And>s. s \<in> init (pnet np p2) \<Longrightarrow> net_ips s = net_tree_ips p2"
and "s \<in> init (pnet np (p1 \<parallel> p2))"
from this(3) obtain s1 s2 where "s1 \<in> init (pnet np p1)"
and "s2 \<in> init (pnet np p2)"
and "s = SubnetS s1 s2" by auto
from this(1-2) have "net_ips s1 = net_tree_ips p1"
and "net_ips s2 = net_tree_ips p2"
using IH1 IH2 by auto
with \<open>s = SubnetS s1 s2\<close> show "net_ips s = net_tree_ips (p1 \<parallel> p2)" by auto
qed
next
fix s a s'
assume "(s, a, s') \<in> trans (pnet np p)"
and "net_ips s = net_tree_ips p"
from this(1) have "net_ips s = net_ips s'"
by (rule pnet_maintains_dom)
with \<open>net_ips s = net_tree_ips p\<close> show "net_ips s' = net_tree_ips p"
by simp
qed
lemma pnet_init_net_ips_net_tree_ips:
assumes "s \<in> init (pnet np p)"
shows "net_ips s = net_tree_ips p"
using assms(1) by (rule reachable_init [THEN pnet_net_ips_net_tree_ips])
lemma pnet_init_in_net_ips_in_net_tree_ips [elim]:
assumes "s \<in> init (pnet np p)"
and "i \<in> net_ips s"
shows "i \<in> net_tree_ips p"
using assms by (clarsimp dest!: pnet_init_net_ips_net_tree_ips)
lemma pnet_init_in_net_tree_ips_in_net_ips [elim]:
assumes "s \<in> init (pnet np p)"
and "i \<in> net_tree_ips p"
shows "i \<in> net_ips s"
using assms by (clarsimp dest!: pnet_init_net_ips_net_tree_ips)
lemma pnet_init_not_in_net_tree_ips_not_in_net_ips [elim]:
assumes "s \<in> init (pnet np p)"
and "i \<notin> net_tree_ips p"
shows "i \<notin> net_ips s"
proof
assume "i \<in> net_ips s"
with assms(1) have "i \<in> net_tree_ips p" ..
with assms(2) show False ..
qed
lemma net_node_reachable_is_node:
assumes "st \<in> reachable (pnet np \<langle>ii; R\<^sub>i\<rangle>) I"
shows "\<exists>ns R. st = NodeS ii ns R"
using assms proof induct
fix s
assume "s \<in> init (pnet np \<langle>ii; R\<^sub>i\<rangle>)"
thus "\<exists>ns R. s = NodeS ii ns R"
by (rule pnet_node_init') simp
next
fix s a s'
assume "s \<in> reachable (pnet np \<langle>ii; R\<^sub>i\<rangle>) I"
and "\<exists>ns R. s = NodeS ii ns R"
and "(s, a, s') \<in> trans (pnet np \<langle>ii; R\<^sub>i\<rangle>)"
and "I a"
thus "\<exists>ns R. s' = NodeS ii ns R"
by (auto simp add: trans_node_comp dest!: node_sos_dest)
qed
lemma partial_net_preserves_subnets:
assumes "(SubnetS s t, a, st') \<in> pnet_sos (trans (pnet np p1)) (trans (pnet np p2))"
shows "\<exists>s' t'. st' = SubnetS s' t'"
using assms by cases simp_all
lemma net_par_reachable_is_subnet:
assumes "st \<in> reachable (pnet np (p1 \<parallel> p2)) I"
shows "\<exists>s t. st = SubnetS s t"
using assms by induct (auto dest!: partial_net_preserves_subnets)
lemma reachable_par_subnet_induct [consumes, case_names init step]:
assumes "SubnetS s t \<in> reachable (pnet np (p1 \<parallel> p2)) I"
and init: "\<And>s t. SubnetS s t \<in> init (pnet np (p1 \<parallel> p2)) \<Longrightarrow> P s t"
and step: "\<And>s t s' t' a. \<lbrakk>
SubnetS s t \<in> reachable (pnet np (p1 \<parallel> p2)) I;
P s t; (SubnetS s t, a, SubnetS s' t') \<in> (trans (pnet np (p1 \<parallel> p2))); I a \<rbrakk>
\<Longrightarrow> P s' t'"
shows "P s t"
using assms(1) proof (induction "SubnetS s t" arbitrary: s t)
fix s t
assume "SubnetS s t \<in> init (pnet np (p1 \<parallel> p2))"
with init show "P s t" .
next
fix st a s' t'
assume "st \<in> reachable (pnet np (p1 \<parallel> p2)) I"
and tr: "(st, a, SubnetS s' t') \<in> trans (pnet np (p1 \<parallel> p2))"
and "I a"
and IH: "\<And>s t. st = SubnetS s t \<Longrightarrow> P s t"
from this(1) obtain s t where "st = SubnetS s t"
and str: "SubnetS s t \<in> reachable (pnet np (p1 \<parallel> p2)) I"
by (metis net_par_reachable_is_subnet)
note this(2)
moreover from IH and \<open>st = SubnetS s t\<close> have "P s t" .
moreover from \<open>st = SubnetS s t\<close> and tr
have "(SubnetS s t, a, SubnetS s' t') \<in> trans (pnet np (p1 \<parallel> p2))" by simp
ultimately show "P s' t'"
using \<open>I a\<close> by (rule step)
qed
lemma subnet_reachable:
assumes "SubnetS s1 s2 \<in> reachable (pnet np (p1 \<parallel> p2)) TT"
shows "s1 \<in> reachable (pnet np p1) TT"
"s2 \<in> reachable (pnet np p2) TT"
proof -
from assms have "s1 \<in> reachable (pnet np p1) TT
\<and> s2 \<in> reachable (pnet np p2) TT"
proof (induction rule: reachable_par_subnet_induct)
fix s1 s2
assume "SubnetS s1 s2 \<in> init (pnet np (p1 \<parallel> p2))"
thus "s1 \<in> reachable (pnet np p1) TT
\<and> s2 \<in> reachable (pnet np p2) TT"
by (auto dest: reachable_init)
next
case (step s1 s2 s1' s2' a)
hence "SubnetS s1 s2 \<in> reachable (pnet np (p1 \<parallel> p2)) TT"
and sr1: "s1 \<in> reachable (pnet np p1) TT"
and sr2: "s2 \<in> reachable (pnet np p2) TT"
and "(SubnetS s1 s2, a, SubnetS s1' s2') \<in> trans (pnet np (p1 \<parallel> p2))" by auto
from this(4)
have "(SubnetS s1 s2, a, SubnetS s1' s2') \<in> pnet_sos (trans (pnet np p1)) (trans (pnet np p2))"
by simp
thus "s1' \<in> reachable (pnet np p1) TT
\<and> s2' \<in> reachable (pnet np p2) TT"
by cases (insert sr1 sr2, auto elim: reachable_step)
qed
thus "s1 \<in> reachable (pnet np p1) TT"
"s2 \<in> reachable (pnet np p2) TT" by auto
qed
lemma delivered_to_node [elim]:
assumes "s \<in> reachable (pnet np \<langle>ii; R\<^sub>i\<rangle>) TT"
and "(s, i:deliver(d), s') \<in> trans (pnet np \<langle>ii; R\<^sub>i\<rangle>)"
shows "i = ii"
proof -
from assms(1) obtain P R where "s = NodeS ii P R"
by (metis net_node_reachable_is_node)
with assms(2) show "i = ii"
by (clarsimp simp add: trans_node_comp elim!: node_deliverTE')
qed
lemma delivered_to_net_ips:
assumes "s \<in> reachable (pnet np p) TT"
and "(s, i:deliver(d), s') \<in> trans (pnet np p)"
shows "i \<in> net_ips s"
using assms proof (induction p arbitrary: s s')
fix ii R\<^sub>i s s'
assume sr: "s \<in> reachable (pnet np \<langle>ii; R\<^sub>i\<rangle>) TT"
and "(s, i:deliver(d), s') \<in> trans (pnet np \<langle>ii; R\<^sub>i\<rangle>)"
from this(2) have tr: "(s, i:deliver(d), s') \<in> node_sos (trans (np ii))" by simp
from sr obtain P R where [simp]: "s = NodeS ii P R"
by (metis net_node_reachable_is_node)
moreover from tr obtain P' R' where [simp]: "s' = NodeS ii P' R'"
by simp (metis node_sos_dest)
ultimately have "i = ii" using tr by auto
thus "i \<in> net_ips s" by simp
next
fix p1 p2 s s'
assume IH1: "\<And>s s'. \<lbrakk> s \<in> reachable (pnet np p1) TT;
(s, i:deliver(d), s') \<in> trans (pnet np p1) \<rbrakk> \<Longrightarrow> i \<in> net_ips s"
and IH2: "\<And>s s'. \<lbrakk> s \<in> reachable (pnet np p2) TT;
(s, i:deliver(d), s') \<in> trans (pnet np p2) \<rbrakk> \<Longrightarrow> i \<in> net_ips s"
and sr: "s \<in> reachable (pnet np (p1 \<parallel> p2)) TT"
and tr: "(s, i:deliver(d), s') \<in> trans (pnet np (p1 \<parallel> p2))"
from tr have "(s, i:deliver(d), s') \<in> pnet_sos (trans (pnet np p1)) (trans (pnet np p2))"
by simp
thus "i \<in> net_ips s"
proof (rule partial_deliverTE)
fix s1 s1' s2
assume "s = SubnetS s1 s2"
and "s' = SubnetS s1' s2"
and tr: "(s1, i:deliver(d), s1') \<in> trans (pnet np p1)"
from sr have "s1 \<in> reachable (pnet np p1) TT"
by (auto simp only: \<open>s = SubnetS s1 s2\<close> elim: subnet_reachable)
hence "i \<in> net_ips s1" using tr by (rule IH1)
thus "i \<in> net_ips s" by (simp add: \<open>s = SubnetS s1 s2\<close>)
next
fix s2 s2' s1
assume "s = SubnetS s1 s2"
and "s' = SubnetS s1 s2'"
and tr: "(s2, i:deliver(d), s2') \<in> trans (pnet np p2)"
from sr have "s2 \<in> reachable (pnet np p2) TT"
by (auto simp only: \<open>s = SubnetS s1 s2\<close> elim: subnet_reachable)
hence "i \<in> net_ips s2" using tr by (rule IH2)
thus "i \<in> net_ips s" by (simp add: \<open>s = SubnetS s1 s2\<close>)
qed
qed
lemma wf_net_tree_net_ips_disjoint [elim]:
assumes "wf_net_tree (p1 \<parallel> p2)"
and "s1 \<in> reachable (pnet np p1) S"
and "s2 \<in> reachable (pnet np p2) S"
shows "net_ips s1 \<inter> net_ips s2 = {}"
proof -
from \<open>wf_net_tree (p1 \<parallel> p2)\<close> have "net_tree_ips p1 \<inter> net_tree_ips p2 = {}" by auto
moreover from assms(2) have "net_ips s1 = net_tree_ips p1" ..
moreover from assms(3) have "net_ips s2 = net_tree_ips p2" ..
ultimately show ?thesis by simp
qed
lemma init_mapstate_Some_aodv_init [elim]:
assumes "s \<in> init (pnet np p)"
and "netmap s i = Some v"
shows "v \<in> init (np i)"
using assms proof (induction p arbitrary: s)
fix ii R s
assume "s \<in> init (pnet np \<langle>ii; R\<rangle>)"
and "netmap s i = Some v"
from this(1) obtain ns where s: "s = NodeS ii ns R"
and ns: "ns \<in> init (np ii)" ..
from s and \<open>netmap s i = Some v\<close> have "i = ii"
by simp (metis domI domIff)
with s ns show "v \<in> init (np i)"
using \<open>netmap s i = Some v\<close> by simp
next
fix p1 p2 s
assume IH1: "\<And>s. s \<in> init (pnet np p1) \<Longrightarrow> netmap s i = Some v \<Longrightarrow> v \<in> init (np i)"
and IH2: "\<And>s. s \<in> init (pnet np p2) \<Longrightarrow> netmap s i = Some v \<Longrightarrow> v \<in> init (np i)"
and "s \<in> init (pnet np (p1 \<parallel> p2))"
and "netmap s i = Some v"
from this(3) obtain s1 s2 where "s = SubnetS s1 s2"
and "s1 \<in> init (pnet np p1)"
and "s2 \<in> init (pnet np p2)" by auto
from this(1) and \<open>netmap s i = Some v\<close>
have "netmap s1 i = Some v \<or> netmap s2 i = Some v" by auto
thus "v \<in> init (np i)"
proof
assume "netmap s1 i = Some v"
with \<open>s1 \<in> init (pnet np p1)\<close> show ?thesis by (rule IH1)
next
assume "netmap s2 i = Some v"
with \<open>s2 \<in> init (pnet np p2)\<close> show ?thesis by (rule IH2)
qed
qed
lemma reachable_connect_netmap [elim]:
assumes "s \<in> reachable (pnet np n) TT"
and "(s, connect(i, i'), s') \<in> trans (pnet np n)"
shows "netmap s' = netmap s"
using assms proof (induction n arbitrary: s s')
fix ii R\<^sub>i s s'
assume sr: "s \<in> reachable (pnet np \<langle>ii; R\<^sub>i\<rangle>) TT"
and "(s, connect(i, i'), s') \<in> trans (pnet np \<langle>ii; R\<^sub>i\<rangle>)"
from this(2) have tr: "(s, connect(i, i'), s') \<in> node_sos (trans (np ii))" ..
from sr obtain p R where "s = NodeS ii p R"
by (metis net_node_reachable_is_node)
with tr show "netmap s' = netmap s"
by (auto elim!: node_sos.cases)
next
fix p1 p2 s s'
assume IH1: "\<And>s s'. \<lbrakk> s \<in> reachable (pnet np p1) TT;
(s, connect(i, i'), s') \<in> trans (pnet np p1) \<rbrakk> \<Longrightarrow> netmap s' = netmap s"
and IH2: "\<And>s s'. \<lbrakk> s \<in> reachable (pnet np p2) TT;
(s, connect(i, i'), s') \<in> trans (pnet np p2) \<rbrakk> \<Longrightarrow> netmap s' = netmap s"
and sr: "s \<in> reachable (pnet np (p1 \<parallel> p2)) TT"
and tr: "(s, connect(i, i'), s') \<in> trans (pnet np (p1 \<parallel> p2))"
from tr have "(s, connect(i, i'), s') \<in> pnet_sos (trans (pnet np p1)) (trans (pnet np p2))"
by simp
thus "netmap s' = netmap s"
proof cases
fix s1 s1' s2 s2'
assume "s = SubnetS s1 s2"
and "s' = SubnetS s1' s2'"
and tr1: "(s1, connect(i, i'), s1') \<in> trans (pnet np p1)"
and tr2: "(s2, connect(i, i'), s2') \<in> trans (pnet np p2)"
from this(1) and sr
have "SubnetS s1 s2 \<in> reachable (pnet np (p1 \<parallel> p2)) TT" by simp
hence sr1: "s1 \<in> reachable (pnet np p1) TT"
and sr2: "s2 \<in> reachable (pnet np p2) TT"
by (auto intro: subnet_reachable)
from sr1 tr1 have "netmap s1' = netmap s1" by (rule IH1)
moreover from sr2 tr2 have "netmap s2' = netmap s2" by (rule IH2)
ultimately show "netmap s' = netmap s"
using \<open>s = SubnetS s1 s2\<close> and \<open>s' = SubnetS s1' s2'\<close> by simp
qed simp_all
qed
lemma reachable_disconnect_netmap [elim]:
assumes "s \<in> reachable (pnet np n) TT"
and "(s, disconnect(i, i'), s') \<in> trans (pnet np n)"
shows "netmap s' = netmap s"
using assms proof (induction n arbitrary: s s')
fix ii R\<^sub>i s s'
assume sr: "s \<in> reachable (pnet np \<langle>ii; R\<^sub>i\<rangle>) TT"
and "(s, disconnect(i, i'), s') \<in> trans (pnet np \<langle>ii; R\<^sub>i\<rangle>)"
from this(2) have tr: "(s, disconnect(i, i'), s') \<in> node_sos (trans (np ii))" ..
from sr obtain p R where "s = NodeS ii p R"
by (metis net_node_reachable_is_node)
with tr show "netmap s' = netmap s"
by (auto elim!: node_sos.cases)
next
fix p1 p2 s s'
assume IH1: "\<And>s s'. \<lbrakk> s \<in> reachable (pnet np p1) TT;
(s, disconnect(i, i'), s') \<in> trans (pnet np p1) \<rbrakk> \<Longrightarrow> netmap s' = netmap s"
and IH2: "\<And>s s'. \<lbrakk> s \<in> reachable (pnet np p2) TT;
(s, disconnect(i, i'), s') \<in> trans (pnet np p2) \<rbrakk> \<Longrightarrow> netmap s' = netmap s"
and sr: "s \<in> reachable (pnet np (p1 \<parallel> p2)) TT"
and tr: "(s, disconnect(i, i'), s') \<in> trans (pnet np (p1 \<parallel> p2))"
from tr have "(s, disconnect(i, i'), s') \<in> pnet_sos (trans (pnet np p1)) (trans (pnet np p2))"
by simp
thus "netmap s' = netmap s"
proof cases
fix s1 s1' s2 s2'
assume "s = SubnetS s1 s2"
and "s' = SubnetS s1' s2'"
and tr1: "(s1, disconnect(i, i'), s1') \<in> trans (pnet np p1)"
and tr2: "(s2, disconnect(i, i'), s2') \<in> trans (pnet np p2)"
from this(1) and sr
have "SubnetS s1 s2 \<in> reachable (pnet np (p1 \<parallel> p2)) TT" by simp
hence sr1: "s1 \<in> reachable (pnet np p1) TT"
and sr2: "s2 \<in> reachable (pnet np p2) TT"
by (auto intro: subnet_reachable)
from sr1 tr1 have "netmap s1' = netmap s1" by (rule IH1)
moreover from sr2 tr2 have "netmap s2' = netmap s2" by (rule IH2)
ultimately show "netmap s' = netmap s"
using \<open>s = SubnetS s1 s2\<close> and \<open>s' = SubnetS s1' s2'\<close> by simp
qed simp_all
qed
fun net_ip_action :: "(ip \<Rightarrow> ('s, 'm seq_action) automaton)
\<Rightarrow> 'm node_action \<Rightarrow> ip \<Rightarrow> net_tree \<Rightarrow> 's net_state \<Rightarrow> 's net_state \<Rightarrow> bool"
where
"net_ip_action np a i (p1 \<parallel> p2) (SubnetS s1 s2) (SubnetS s1' s2') =
((i \<in> net_ips s1 \<longrightarrow> ((s1, a, s1') \<in> trans (pnet np p1)
\<and> s2' = s2 \<and> net_ip_action np a i p1 s1 s1'))
\<and> (i \<in> net_ips s2 \<longrightarrow> ((s2, a, s2') \<in> trans (pnet np p2))
\<and> s1' = s1 \<and> net_ip_action np a i p2 s2 s2'))"
| "net_ip_action np a i p s s' = True"
lemma pnet_tau_single_node [elim]:
assumes "wf_net_tree p"
and "s \<in> reachable (pnet np p) TT"
and "(s, \<tau>, s') \<in> trans (pnet np p)"
shows "\<exists>i\<in>net_ips s. ((\<forall>j. j\<noteq>i \<longrightarrow> netmap s' j = netmap s j)
\<and> net_ip_action np \<tau> i p s s')"
using assms proof (induction p arbitrary: s s')
fix ii R\<^sub>i s s'
assume "s \<in> reachable (pnet np \<langle>ii; R\<^sub>i\<rangle>) TT"
and "(s, \<tau>, s') \<in> trans (pnet np \<langle>ii; R\<^sub>i\<rangle>)"
from this obtain p R p' R' where "s = NodeS ii p R" and "s' = NodeS ii p' R'"
by (metis (opaque_lifting, no_types) TT_True net_node_reachable_is_node
reachable_step)
hence "net_ips s = {ii}"
and "net_ips s' = {ii}" by simp_all
hence "\<exists>i\<in>dom (netmap s). \<forall>j. j \<noteq> i \<longrightarrow> netmap s' j = netmap s j"
by (simp add: net_ips_is_dom_netmap)
thus "\<exists>i\<in>net_ips s. (\<forall>j. j \<noteq> i \<longrightarrow> netmap s' j = netmap s j)
\<and> net_ip_action np \<tau> i (\<langle>ii; R\<^sub>i\<rangle>) s s'"
by (simp add: net_ips_is_dom_netmap)
next
fix p1 p2 s s'
assume IH1: "\<And>s s'. \<lbrakk> wf_net_tree p1;
s \<in> reachable (pnet np p1) TT;
(s, \<tau>, s') \<in> trans (pnet np p1) \<rbrakk>
\<Longrightarrow> \<exists>i\<in>net_ips s. (\<forall>j. j \<noteq> i \<longrightarrow> netmap s' j = netmap s j)
\<and> net_ip_action np \<tau> i p1 s s'"
and IH2: "\<And>s s'. \<lbrakk> wf_net_tree p2;
s \<in> reachable (pnet np p2) TT;
(s, \<tau>, s') \<in> trans (pnet np p2) \<rbrakk>
\<Longrightarrow> \<exists>i\<in>net_ips s. (\<forall>j. j \<noteq> i \<longrightarrow> netmap s' j = netmap s j)
\<and> net_ip_action np \<tau> i p2 s s'"
and sr: "s \<in> reachable (pnet np (p1 \<parallel> p2)) TT"
and "wf_net_tree (p1 \<parallel> p2)"
and tr: "(s, \<tau>, s') \<in> trans (pnet np (p1 \<parallel> p2))"
from \<open>wf_net_tree (p1 \<parallel> p2)\<close> have "net_tree_ips p1 \<inter> net_tree_ips p2 = {}"
and "wf_net_tree p1"
and "wf_net_tree p2" by auto
from tr have "(s, \<tau>, s') \<in> pnet_sos (trans (pnet np p1)) (trans (pnet np p2))" by simp
thus "\<exists>i\<in>net_ips s. (\<forall>j. j \<noteq> i \<longrightarrow> netmap s' j = netmap s j)
\<and> net_ip_action np \<tau> i (p1 \<parallel> p2) s s'"
proof cases
fix s1 s1' s2
assume subs: "s = SubnetS s1 s2"
and subs': "s' = SubnetS s1' s2"
and tr1: "(s1, \<tau>, s1') \<in> trans (pnet np p1)"
from sr have sr1: "s1 \<in> reachable (pnet np p1) TT"
and "s2 \<in> reachable (pnet np p2) TT"
by (simp_all only: subs) (erule subnet_reachable)+
with \<open>net_tree_ips p1 \<inter> net_tree_ips p2 = {}\<close> have "dom(netmap s1) \<inter> dom(netmap s2) = {}"
by (metis net_ips_is_dom_netmap pnet_net_ips_net_tree_ips)
from \<open>wf_net_tree p1\<close> sr1 tr1 obtain i where "i\<in>dom(netmap s1)"
and *: "\<forall>j. j \<noteq> i \<longrightarrow> netmap s1' j = netmap s1 j"
and "net_ip_action np \<tau> i p1 s1 s1'"
by (auto simp add: net_ips_is_dom_netmap dest!: IH1)
from this(1) and \<open>dom(netmap s1) \<inter> dom(netmap s2) = {}\<close> have "i\<notin>dom(netmap s2)"
by auto
with subs subs' tr1 \<open>net_ip_action np \<tau> i p1 s1 s1'\<close> have "net_ip_action np \<tau> i (p1 \<parallel> p2) s s'"
by (simp add: net_ips_is_dom_netmap)
moreover have "\<forall>j. j \<noteq> i \<longrightarrow> (netmap s1' ++ netmap s2) j = (netmap s1 ++ netmap s2) j"
proof (intro allI impI)
fix j
assume "j \<noteq> i"
with * have "netmap s1' j = netmap s1 j" by simp
thus "(netmap s1' ++ netmap s2) j = (netmap s1 ++ netmap s2) j"
by (metis (opaque_lifting, mono_tags) map_add_dom_app_simps(1) map_add_dom_app_simps(3))
qed
ultimately show ?thesis using \<open>i\<in>dom(netmap s1)\<close> subs subs'
by (auto simp add: net_ips_is_dom_netmap)
next
fix s2 s2' s1
assume subs: "s = SubnetS s1 s2"
and subs': "s' = SubnetS s1 s2'"
and tr2: "(s2, \<tau>, s2') \<in> trans (pnet np p2)"
from sr have "s1 \<in> reachable (pnet np p1) TT"
and sr2: "s2 \<in> reachable (pnet np p2) TT"
by (simp_all only: subs) (erule subnet_reachable)+
with \<open>net_tree_ips p1 \<inter> net_tree_ips p2 = {}\<close> have "dom(netmap s1) \<inter> dom(netmap s2) = {}"
by (metis net_ips_is_dom_netmap pnet_net_ips_net_tree_ips)
from \<open>wf_net_tree p2\<close> sr2 tr2 obtain i where "i\<in>dom(netmap s2)"
and *: "\<forall>j. j \<noteq> i \<longrightarrow> netmap s2' j = netmap s2 j"
and "net_ip_action np \<tau> i p2 s2 s2'"
by (auto simp add: net_ips_is_dom_netmap dest!: IH2)
from this(1) and \<open>dom(netmap s1) \<inter> dom(netmap s2) = {}\<close> have "i\<notin>dom(netmap s1)"
by auto
with subs subs' tr2 \<open>net_ip_action np \<tau> i p2 s2 s2'\<close> have "net_ip_action np \<tau> i (p1 \<parallel> p2) s s'"
by (simp add: net_ips_is_dom_netmap)
moreover have "\<forall>j. j \<noteq> i \<longrightarrow> (netmap s1 ++ netmap s2') j = (netmap s1 ++ netmap s2) j"
proof (intro allI impI)
fix j
assume "j \<noteq> i"
with * have "netmap s2' j = netmap s2 j" by simp
thus "(netmap s1 ++ netmap s2') j = (netmap s1 ++ netmap s2) j"
by (metis (opaque_lifting, mono_tags) domD map_add_Some_iff map_add_dom_app_simps(3))
qed
ultimately show ?thesis using \<open>i\<in>dom(netmap s2)\<close> subs subs'
by (clarsimp simp add: net_ips_is_dom_netmap)
(metis domI dom_map_add map_add_find_right)
qed simp_all
qed
lemma pnet_deliver_single_node [elim]:
assumes "wf_net_tree p"
and "s \<in> reachable (pnet np p) TT"
and "(s, i:deliver(d), s') \<in> trans (pnet np p)"
shows "(\<forall>j. j\<noteq>i \<longrightarrow> netmap s' j = netmap s j) \<and> net_ip_action np (i:deliver(d)) i p s s'"
(is "?P p s s'")
using assms proof (induction p arbitrary: s s')
fix ii R\<^sub>i s s'
assume sr: "s \<in> reachable (pnet np \<langle>ii; R\<^sub>i\<rangle>) TT"
and tr: "(s, i:deliver(d), s') \<in> trans (pnet np \<langle>ii; R\<^sub>i\<rangle>)"
from this obtain p R p' R' where "s = NodeS ii p R" and "s' = NodeS ii p' R'"
by (metis (opaque_lifting, no_types) TT_True net_node_reachable_is_node
reachable_step)
hence "net_ips s = {ii}"
and "net_ips s' = {ii}" by simp_all
hence "\<forall>j. j \<noteq> ii \<longrightarrow> netmap s' j = netmap s j"
by simp
moreover from sr tr have "i = ii" by (rule delivered_to_node)
ultimately show "(\<forall>j. j \<noteq> i \<longrightarrow> netmap s' j = netmap s j)
\<and> net_ip_action np (i:deliver(d)) i (\<langle>ii; R\<^sub>i\<rangle>) s s'"
by simp
next
fix p1 p2 s s'
assume IH1: "\<And>s s'. \<lbrakk> wf_net_tree p1;
s \<in> reachable (pnet np p1) TT;
(s, i:deliver(d), s') \<in> trans (pnet np p1) \<rbrakk>
\<Longrightarrow> (\<forall>j. j \<noteq> i \<longrightarrow> netmap s' j = netmap s j)
\<and> net_ip_action np (i:deliver(d)) i p1 s s'"
and IH2: "\<And>s s'. \<lbrakk> wf_net_tree p2;
s \<in> reachable (pnet np p2) TT;
(s, i:deliver(d), s') \<in> trans (pnet np p2) \<rbrakk>
\<Longrightarrow> (\<forall>j. j \<noteq> i \<longrightarrow> netmap s' j = netmap s j)
\<and> net_ip_action np (i:deliver(d)) i p2 s s'"
and sr: "s \<in> reachable (pnet np (p1 \<parallel> p2)) TT"
and "wf_net_tree (p1 \<parallel> p2)"
and tr: "(s, i:deliver(d), s') \<in> trans (pnet np (p1 \<parallel> p2))"
from \<open>wf_net_tree (p1 \<parallel> p2)\<close> have "net_tree_ips p1 \<inter> net_tree_ips p2 = {}"
and "wf_net_tree p1"
and "wf_net_tree p2" by auto
from tr have "(s, i:deliver(d), s') \<in> pnet_sos (trans (pnet np p1)) (trans (pnet np p2))" by simp
thus "(\<forall>j. j \<noteq> i \<longrightarrow> netmap s' j = netmap s j)
\<and> net_ip_action np (i:deliver(d)) i (p1 \<parallel> p2) s s'"
proof cases
fix s1 s1' s2
assume subs: "s = SubnetS s1 s2"
and subs': "s' = SubnetS s1' s2"
and tr1: "(s1, i:deliver(d), s1') \<in> trans (pnet np p1)"
from sr have sr1: "s1 \<in> reachable (pnet np p1) TT"
and "s2 \<in> reachable (pnet np p2) TT"
by (simp_all only: subs) (erule subnet_reachable)+
with \<open>net_tree_ips p1 \<inter> net_tree_ips p2 = {}\<close> have "dom(netmap s1) \<inter> dom(netmap s2) = {}"
by (metis net_ips_is_dom_netmap pnet_net_ips_net_tree_ips)
moreover from sr1 tr1 have "i \<in> net_ips s1" by (rule delivered_to_net_ips)
ultimately have "i\<notin>dom(netmap s2)" by (auto simp add: net_ips_is_dom_netmap)
from \<open>wf_net_tree p1\<close> sr1 tr1 have *: "\<forall>j. j \<noteq> i \<longrightarrow> netmap s1' j = netmap s1 j"
and "net_ip_action np (i:deliver(d)) i p1 s1 s1'"
by (auto dest!: IH1)
from subs subs' tr1 this(2) \<open>i\<notin>dom(netmap s2)\<close>
have "net_ip_action np (i:deliver(d)) i (p1 \<parallel> p2) s s'"
by (simp add: net_ips_is_dom_netmap)
moreover have "\<forall>j. j \<noteq> i \<longrightarrow> (netmap s1' ++ netmap s2) j = (netmap s1 ++ netmap s2) j"
proof (intro allI impI)
fix j
assume "j \<noteq> i"
with * have "netmap s1' j = netmap s1 j" by simp
thus "(netmap s1' ++ netmap s2) j = (netmap s1 ++ netmap s2) j"
by (metis (opaque_lifting, mono_tags) map_add_dom_app_simps(1) map_add_dom_app_simps(3))
qed
ultimately show ?thesis using \<open>i\<in>net_ips s1\<close> subs subs' by auto
next
fix s2 s2' s1
assume subs: "s = SubnetS s1 s2"
and subs': "s' = SubnetS s1 s2'"
and tr2: "(s2, i:deliver(d), s2') \<in> trans (pnet np p2)"
from sr have "s1 \<in> reachable (pnet np p1) TT"
and sr2: "s2 \<in> reachable (pnet np p2) TT"
by (simp_all only: subs) (erule subnet_reachable)+
with \<open>net_tree_ips p1 \<inter> net_tree_ips p2 = {}\<close> have "dom(netmap s1) \<inter> dom(netmap s2) = {}"
by (metis net_ips_is_dom_netmap pnet_net_ips_net_tree_ips)
moreover from sr2 tr2 have "i \<in> net_ips s2" by (rule delivered_to_net_ips)
ultimately have "i\<notin>dom(netmap s1)" by (auto simp add: net_ips_is_dom_netmap)
from \<open>wf_net_tree p2\<close> sr2 tr2 have *: "\<forall>j. j \<noteq> i \<longrightarrow> netmap s2' j = netmap s2 j"
and "net_ip_action np (i:deliver(d)) i p2 s2 s2'"
by (auto dest!: IH2)
from subs subs' tr2 this(2) \<open>i\<notin>dom(netmap s1)\<close>
have "net_ip_action np (i:deliver(d)) i (p1 \<parallel> p2) s s'"
by (simp add: net_ips_is_dom_netmap)
moreover have "\<forall>j. j \<noteq> i \<longrightarrow> (netmap s1 ++ netmap s2') j = (netmap s1 ++ netmap s2) j"
proof (intro allI impI)
fix j
assume "j \<noteq> i"
with * have "netmap s2' j = netmap s2 j" by simp
thus "(netmap s1 ++ netmap s2') j = (netmap s1 ++ netmap s2) j"
by (metis (opaque_lifting, mono_tags) domD map_add_Some_iff map_add_dom_app_simps(3))
qed
ultimately show ?thesis using \<open>i\<in>net_ips s2\<close> subs subs' by auto
qed simp_all
qed
end
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