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(* Title: Loop_Freedom.thy
License: BSD 2-Clause. See LICENSE.
Author: Timothy Bourke, Inria
*)
section "Routing graphs and loop freedom"
theory Loop_Freedom
imports Aodv_Predicates Fresher
begin
text \<open>Define the central theorem that relates an invariant over network states to the absence
of loops in the associate routing graph.\<close>
definition
rt_graph :: "(ip \<Rightarrow> state) \<Rightarrow> ip \<Rightarrow> ip rel"
where
"rt_graph \<sigma> = (\<lambda>dip.
{(ip, ip') | ip ip' dsn dsk hops pre.
ip \<noteq> dip \<and> rt (\<sigma> ip) dip = Some (dsn, dsk, val, hops, ip', pre)})"
text \<open>Given the state of a network @{term \<sigma>}, a routing graph for a given destination
ip address @{term dip} abstracts the details of routing tables into nodes
(ip addresses) and vertices (valid routes between ip addresses).\<close>
lemma rt_graphE [elim]:
fixes n dip ip ip'
assumes "(ip, ip') \<in> rt_graph \<sigma> dip"
shows "ip \<noteq> dip \<and> (\<exists>r. rt (\<sigma> ip) = r
\<and> (\<exists>dsn dsk hops pre. r dip = Some (dsn, dsk, val, hops, ip', pre)))"
using assms unfolding rt_graph_def by auto
lemma rt_graph_vD [dest]:
"\<And>ip ip' \<sigma> dip. (ip, ip') \<in> rt_graph \<sigma> dip \<Longrightarrow> dip \<in> vD(rt (\<sigma> ip))"
unfolding rt_graph_def vD_def by auto
lemma rt_graph_vD_trans [dest]:
"\<And>ip ip' \<sigma> dip. (ip, ip') \<in> (rt_graph \<sigma> dip)\<^sup>+ \<Longrightarrow> dip \<in> vD(rt (\<sigma> ip))"
by (erule converse_tranclE) auto
lemma rt_graph_not_dip [dest]:
"\<And>ip ip' \<sigma> dip. (ip, ip') \<in> rt_graph \<sigma> dip \<Longrightarrow> ip \<noteq> dip"
unfolding rt_graph_def by auto
lemma rt_graph_not_dip_trans [dest]:
"\<And>ip ip' \<sigma> dip. (ip, ip') \<in> (rt_graph \<sigma> dip)\<^sup>+ \<Longrightarrow> ip \<noteq> dip"
by (erule converse_tranclE) auto
text "NB: the property below cannot be lifted to the transitive closure"
lemma rt_graph_nhip_is_nhop [dest]:
"\<And>ip ip' \<sigma> dip. (ip, ip') \<in> rt_graph \<sigma> dip \<Longrightarrow> ip' = the (nhop (rt (\<sigma> ip)) dip)"
unfolding rt_graph_def by auto
theorem inv_to_loop_freedom:
assumes "\<forall>i dip. let nhip = the (nhop (rt (\<sigma> i)) dip)
in dip \<in> vD (rt (\<sigma> i)) \<inter> vD (rt (\<sigma> nhip)) \<and> nhip \<noteq> dip
\<longrightarrow> (rt (\<sigma> i)) \<sqsubset>\<^bsub>dip\<^esub> (rt (\<sigma> nhip))"
shows "\<forall>dip. irrefl ((rt_graph \<sigma> dip)\<^sup>+)"
using assms proof (intro allI)
fix \<sigma> :: "ip \<Rightarrow> state" and dip
assume inv: "\<forall>ip dip.
let nhip = the (nhop (rt (\<sigma> ip)) dip)
in dip \<in> vD(rt (\<sigma> ip)) \<inter> vD(rt (\<sigma> nhip)) \<and>
nhip \<noteq> dip \<longrightarrow> rt (\<sigma> ip) \<sqsubset>\<^bsub>dip\<^esub> rt (\<sigma> nhip)"
{ fix ip ip'
assume "(ip, ip') \<in> (rt_graph \<sigma> dip)\<^sup>+"
and "dip \<in> vD(rt (\<sigma> ip'))"
and "ip' \<noteq> dip"
hence "rt (\<sigma> ip) \<sqsubset>\<^bsub>dip\<^esub> rt (\<sigma> ip')"
proof induction
fix nhip
assume "(ip, nhip) \<in> rt_graph \<sigma> dip"
and "dip \<in> vD(rt (\<sigma> nhip))"
and "nhip \<noteq> dip"
from \<open>(ip, nhip) \<in> rt_graph \<sigma> dip\<close> have "dip \<in> vD(rt (\<sigma> ip))"
and "nhip = the (nhop (rt (\<sigma> ip)) dip)"
by auto
from \<open>dip \<in> vD(rt (\<sigma> ip))\<close> and \<open>dip \<in> vD(rt (\<sigma> nhip))\<close>
have "dip \<in> vD(rt (\<sigma> ip)) \<inter> vD(rt (\<sigma> nhip))" ..
with \<open>nhip = the (nhop (rt (\<sigma> ip)) dip)\<close>
and \<open>nhip \<noteq> dip\<close>
and inv
show "rt (\<sigma> ip) \<sqsubset>\<^bsub>dip\<^esub> rt (\<sigma> nhip)"
by (clarsimp simp: Let_def)
next
fix nhip nhip'
assume "(ip, nhip) \<in> (rt_graph \<sigma> dip)\<^sup>+"
and "(nhip, nhip') \<in> rt_graph \<sigma> dip"
and IH: "\<lbrakk> dip \<in> vD(rt (\<sigma> nhip)); nhip \<noteq> dip \<rbrakk> \<Longrightarrow> rt (\<sigma> ip) \<sqsubset>\<^bsub>dip\<^esub> rt (\<sigma> nhip)"
and "dip \<in> vD(rt (\<sigma> nhip'))"
and "nhip' \<noteq> dip"
from \<open>(nhip, nhip') \<in> rt_graph \<sigma> dip\<close> have 1: "dip \<in> vD(rt (\<sigma> nhip))"
and 2: "nhip \<noteq> dip"
and "nhip' = the (nhop (rt (\<sigma> nhip)) dip)"
by auto
from 1 2 have "rt (\<sigma> ip) \<sqsubset>\<^bsub>dip\<^esub> rt (\<sigma> nhip)" by (rule IH)
also have "rt (\<sigma> nhip) \<sqsubset>\<^bsub>dip\<^esub> rt (\<sigma> nhip')"
proof -
from \<open>dip \<in> vD(rt (\<sigma> nhip))\<close> and \<open>dip \<in> vD(rt (\<sigma> nhip'))\<close>
have "dip \<in> vD(rt (\<sigma> nhip)) \<inter> vD(rt (\<sigma> nhip'))" ..
with \<open>nhip' \<noteq> dip\<close>
and \<open>nhip' = the (nhop (rt (\<sigma> nhip)) dip)\<close>
and inv
show "rt (\<sigma> nhip) \<sqsubset>\<^bsub>dip\<^esub> rt (\<sigma> nhip')"
by (clarsimp simp: Let_def)
qed
finally show "rt (\<sigma> ip) \<sqsubset>\<^bsub>dip\<^esub> rt (\<sigma> nhip')" .
qed } note fresher = this
show "irrefl ((rt_graph \<sigma> dip)\<^sup>+)"
unfolding irrefl_def proof (intro allI notI)
fix ip
assume "(ip, ip) \<in> (rt_graph \<sigma> dip)\<^sup>+"
moreover then have "dip \<in> vD(rt (\<sigma> ip))"
and "ip \<noteq> dip"
by auto
ultimately have "rt (\<sigma> ip) \<sqsubset>\<^bsub>dip\<^esub> rt (\<sigma> ip)" by (rule fresher)
thus False by simp
qed
qed
end
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