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(* Title: Aodv.thy
License: BSD 2-Clause. See LICENSE.
Author: Timothy Bourke, Inria
*)
section "The AODV protocol"
theory Aodv
imports Aodv_Data Aodv_Message
AWN.AWN_SOS_Labels AWN.AWN_Invariants
begin
subsection "Data state"
record state =
ip :: "ip"
sn :: "sqn"
rt :: "rt"
rreqs :: "(ip \<times> rreqid) set"
store :: "store"
(* all locals *)
msg :: "msg"
data :: "data"
dests :: "ip \<rightharpoonup> sqn"
pre :: "ip set"
rreqid :: "rreqid"
dip :: "ip"
oip :: "ip"
hops :: "nat"
dsn :: "sqn"
dsk :: "k"
osn :: "sqn"
sip :: "ip"
abbreviation aodv_init :: "ip \<Rightarrow> state"
where "aodv_init i \<equiv> \<lparr>
ip = i,
sn = 1,
rt = Map.empty,
rreqs = {},
store = Map.empty,
msg = (SOME x. True),
data = (SOME x. True),
dests = (SOME x. True),
pre = (SOME x. True),
rreqid = (SOME x. True),
dip = (SOME x. True),
oip = (SOME x. True),
hops = (SOME x. True),
dsn = (SOME x. True),
dsk = (SOME x. True),
osn = (SOME x. True),
sip = (SOME x. x \<noteq> i)
\<rparr>"
lemma some_neq_not_eq [simp]: "\<not>((SOME x :: nat. x \<noteq> i) = i)"
by (subst some_eq_ex) (metis zero_neq_numeral)
definition clear_locals :: "state \<Rightarrow> state"
where "clear_locals \<xi> = \<xi> \<lparr>
msg := (SOME x. True),
data := (SOME x. True),
dests := (SOME x. True),
pre := (SOME x. True),
rreqid := (SOME x. True),
dip := (SOME x. True),
oip := (SOME x. True),
hops := (SOME x. True),
dsn := (SOME x. True),
dsk := (SOME x. True),
osn := (SOME x. True),
sip := (SOME x. x \<noteq> ip \<xi>)
\<rparr>"
lemma clear_locals_sip_not_ip [simp]: "\<not>(sip (clear_locals \<xi>) = ip \<xi>)"
unfolding clear_locals_def by simp
lemma clear_locals_but_not_globals [simp]:
"ip (clear_locals \<xi>) = ip \<xi>"
"sn (clear_locals \<xi>) = sn \<xi>"
"rt (clear_locals \<xi>) = rt \<xi>"
"rreqs (clear_locals \<xi>) = rreqs \<xi>"
"store (clear_locals \<xi>) = store \<xi>"
unfolding clear_locals_def by auto
subsection "Auxilliary message handling definitions"
definition is_newpkt
where "is_newpkt \<xi> \<equiv> case msg \<xi> of
Newpkt data' dip' \<Rightarrow> { \<xi>\<lparr>data := data', dip := dip'\<rparr> }
| _ \<Rightarrow> {}"
definition is_pkt
where "is_pkt \<xi> \<equiv> case msg \<xi> of
Pkt data' dip' oip' \<Rightarrow> { \<xi>\<lparr> data := data', dip := dip', oip := oip' \<rparr> }
| _ \<Rightarrow> {}"
definition is_rreq
where "is_rreq \<xi> \<equiv> case msg \<xi> of
Rreq hops' rreqid' dip' dsn' dsk' oip' osn' sip' \<Rightarrow>
{ \<xi>\<lparr> hops := hops', rreqid := rreqid', dip := dip', dsn := dsn',
dsk := dsk', oip := oip', osn := osn', sip := sip' \<rparr> }
| _ \<Rightarrow> {}"
lemma is_rreq_asm [dest!]:
assumes "\<xi>' \<in> is_rreq \<xi>"
shows "(\<exists>hops' rreqid' dip' dsn' dsk' oip' osn' sip'.
msg \<xi> = Rreq hops' rreqid' dip' dsn' dsk' oip' osn' sip' \<and>
\<xi>' = \<xi>\<lparr> hops := hops', rreqid := rreqid', dip := dip', dsn := dsn',
dsk := dsk', oip := oip', osn := osn', sip := sip' \<rparr>)"
using assms unfolding is_rreq_def
by (cases "msg \<xi>") simp_all
definition is_rrep
where "is_rrep \<xi> \<equiv> case msg \<xi> of
Rrep hops' dip' dsn' oip' sip' \<Rightarrow>
{ \<xi>\<lparr> hops := hops', dip := dip', dsn := dsn', oip := oip', sip := sip' \<rparr> }
| _ \<Rightarrow> {}"
lemma is_rrep_asm [dest!]:
assumes "\<xi>' \<in> is_rrep \<xi>"
shows "(\<exists>hops' dip' dsn' oip' sip'.
msg \<xi> = Rrep hops' dip' dsn' oip' sip' \<and>
\<xi>' = \<xi>\<lparr> hops := hops', dip := dip', dsn := dsn', oip := oip', sip := sip' \<rparr>)"
using assms unfolding is_rrep_def
by (cases "msg \<xi>") simp_all
definition is_rerr
where "is_rerr \<xi> \<equiv> case msg \<xi> of
Rerr dests' sip' \<Rightarrow> { \<xi>\<lparr> dests := dests', sip := sip' \<rparr> }
| _ \<Rightarrow> {}"
lemma is_rerr_asm [dest!]:
assumes "\<xi>' \<in> is_rerr \<xi>"
shows "(\<exists>dests' sip'.
msg \<xi> = Rerr dests' sip' \<and>
\<xi>' = \<xi>\<lparr> dests := dests', sip := sip' \<rparr>)"
using assms unfolding is_rerr_def
by (cases "msg \<xi>") simp_all
lemmas is_msg_defs =
is_rerr_def is_rrep_def is_rreq_def is_pkt_def is_newpkt_def
lemma is_msg_inv_ip [simp]:
"\<xi>' \<in> is_rerr \<xi> \<Longrightarrow> ip \<xi>' = ip \<xi>"
"\<xi>' \<in> is_rrep \<xi> \<Longrightarrow> ip \<xi>' = ip \<xi>"
"\<xi>' \<in> is_rreq \<xi> \<Longrightarrow> ip \<xi>' = ip \<xi>"
"\<xi>' \<in> is_pkt \<xi> \<Longrightarrow> ip \<xi>' = ip \<xi>"
"\<xi>' \<in> is_newpkt \<xi> \<Longrightarrow> ip \<xi>' = ip \<xi>"
unfolding is_msg_defs
by (cases "msg \<xi>", clarsimp+)+
lemma is_msg_inv_sn [simp]:
"\<xi>' \<in> is_rerr \<xi> \<Longrightarrow> sn \<xi>' = sn \<xi>"
"\<xi>' \<in> is_rrep \<xi> \<Longrightarrow> sn \<xi>' = sn \<xi>"
"\<xi>' \<in> is_rreq \<xi> \<Longrightarrow> sn \<xi>' = sn \<xi>"
"\<xi>' \<in> is_pkt \<xi> \<Longrightarrow> sn \<xi>' = sn \<xi>"
"\<xi>' \<in> is_newpkt \<xi> \<Longrightarrow> sn \<xi>' = sn \<xi>"
unfolding is_msg_defs
by (cases "msg \<xi>", clarsimp+)+
lemma is_msg_inv_rt [simp]:
"\<xi>' \<in> is_rerr \<xi> \<Longrightarrow> rt \<xi>' = rt \<xi>"
"\<xi>' \<in> is_rrep \<xi> \<Longrightarrow> rt \<xi>' = rt \<xi>"
"\<xi>' \<in> is_rreq \<xi> \<Longrightarrow> rt \<xi>' = rt \<xi>"
"\<xi>' \<in> is_pkt \<xi> \<Longrightarrow> rt \<xi>' = rt \<xi>"
"\<xi>' \<in> is_newpkt \<xi> \<Longrightarrow> rt \<xi>' = rt \<xi>"
unfolding is_msg_defs
by (cases "msg \<xi>", clarsimp+)+
lemma is_msg_inv_rreqs [simp]:
"\<xi>' \<in> is_rerr \<xi> \<Longrightarrow> rreqs \<xi>' = rreqs \<xi>"
"\<xi>' \<in> is_rrep \<xi> \<Longrightarrow> rreqs \<xi>' = rreqs \<xi>"
"\<xi>' \<in> is_rreq \<xi> \<Longrightarrow> rreqs \<xi>' = rreqs \<xi>"
"\<xi>' \<in> is_pkt \<xi> \<Longrightarrow> rreqs \<xi>' = rreqs \<xi>"
"\<xi>' \<in> is_newpkt \<xi> \<Longrightarrow> rreqs \<xi>' = rreqs \<xi>"
unfolding is_msg_defs
by (cases "msg \<xi>", clarsimp+)+
lemma is_msg_inv_store [simp]:
"\<xi>' \<in> is_rerr \<xi> \<Longrightarrow> store \<xi>' = store \<xi>"
"\<xi>' \<in> is_rrep \<xi> \<Longrightarrow> store \<xi>' = store \<xi>"
"\<xi>' \<in> is_rreq \<xi> \<Longrightarrow> store \<xi>' = store \<xi>"
"\<xi>' \<in> is_pkt \<xi> \<Longrightarrow> store \<xi>' = store \<xi>"
"\<xi>' \<in> is_newpkt \<xi> \<Longrightarrow> store \<xi>' = store \<xi>"
unfolding is_msg_defs
by (cases "msg \<xi>", clarsimp+)+
lemma is_msg_inv_sip [simp]:
"\<xi>' \<in> is_pkt \<xi> \<Longrightarrow> sip \<xi>' = sip \<xi>"
"\<xi>' \<in> is_newpkt \<xi> \<Longrightarrow> sip \<xi>' = sip \<xi>"
unfolding is_msg_defs
by (cases "msg \<xi>", clarsimp+)+
subsection "The protocol process"
datatype pseqp =
PAodv
| PNewPkt
| PPkt
| PRreq
| PRrep
| PRerr
fun nat_of_seqp :: "pseqp \<Rightarrow> nat"
where
"nat_of_seqp PAodv = 1"
| "nat_of_seqp PPkt = 2"
| "nat_of_seqp PNewPkt = 3"
| "nat_of_seqp PRreq = 4"
| "nat_of_seqp PRrep = 5"
| "nat_of_seqp PRerr = 6"
instantiation "pseqp" :: ord
begin
definition less_eq_seqp [iff]: "l1 \<le> l2 = (nat_of_seqp l1 \<le> nat_of_seqp l2)"
definition less_seqp [iff]: "l1 < l2 = (nat_of_seqp l1 < nat_of_seqp l2)"
instance ..
end
abbreviation AODV
where
"AODV \<equiv> \<lambda>_. \<lbrakk>clear_locals\<rbrakk> call(PAodv)"
abbreviation PKT
where
"PKT args \<equiv>
\<lbrakk>\<xi>. let (data, dip, oip) = args \<xi> in
(clear_locals \<xi>) \<lparr> data := data, dip := dip, oip := oip \<rparr>\<rbrakk>
call(PPkt)"
abbreviation NEWPKT
where
"NEWPKT args \<equiv>
\<lbrakk>\<xi>. let (data, dip) = args \<xi> in
(clear_locals \<xi>) \<lparr> data := data, dip := dip \<rparr>\<rbrakk>
call(PNewPkt)"
abbreviation RREQ
where
"RREQ args \<equiv>
\<lbrakk>\<xi>. let (hops, rreqid, dip, dsn, dsk, oip, osn, sip) = args \<xi> in
(clear_locals \<xi>) \<lparr> hops := hops, rreqid := rreqid, dip := dip,
dsn := dsn, dsk := dsk, oip := oip,
osn := osn, sip := sip \<rparr>\<rbrakk>
call(PRreq)"
abbreviation RREP
where
"RREP args \<equiv>
\<lbrakk>\<xi>. let (hops, dip, dsn, oip, sip) = args \<xi> in
(clear_locals \<xi>) \<lparr> hops := hops, dip := dip, dsn := dsn,
oip := oip, sip := sip \<rparr>\<rbrakk>
call(PRrep)"
abbreviation RERR
where
"RERR args \<equiv>
\<lbrakk>\<xi>. let (dests, sip) = args \<xi> in
(clear_locals \<xi>) \<lparr> dests := dests, sip := sip \<rparr>\<rbrakk>
call(PRerr)"
fun \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V :: "(state, msg, pseqp, pseqp label) seqp_env"
where
"\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PAodv = labelled PAodv (
receive(\<lambda>msg' \<xi>. \<xi> \<lparr> msg := msg' \<rparr>).
( \<langle>is_newpkt\<rangle> NEWPKT(\<lambda>\<xi>. (data \<xi>, ip \<xi>))
\<oplus> \<langle>is_pkt\<rangle> PKT(\<lambda>\<xi>. (data \<xi>, dip \<xi>, oip \<xi>))
\<oplus> \<langle>is_rreq\<rangle>
\<lbrakk>\<xi>. \<xi> \<lparr>rt := update (rt \<xi>) (sip \<xi>) (0, unk, val, 1, sip \<xi>, {}) \<rparr>\<rbrakk>
RREQ(\<lambda>\<xi>. (hops \<xi>, rreqid \<xi>, dip \<xi>, dsn \<xi>, dsk \<xi>, oip \<xi>, osn \<xi>, sip \<xi>))
\<oplus> \<langle>is_rrep\<rangle>
\<lbrakk>\<xi>. \<xi> \<lparr>rt := update (rt \<xi>) (sip \<xi>) (0, unk, val, 1, sip \<xi>, {}) \<rparr>\<rbrakk>
RREP(\<lambda>\<xi>. (hops \<xi>, dip \<xi>, dsn \<xi>, oip \<xi>, sip \<xi>))
\<oplus> \<langle>is_rerr\<rangle>
\<lbrakk>\<xi>. \<xi> \<lparr>rt := update (rt \<xi>) (sip \<xi>) (0, unk, val, 1, sip \<xi>, {}) \<rparr>\<rbrakk>
RERR(\<lambda>\<xi>. (dests \<xi>, sip \<xi>))
)
\<oplus> \<langle>\<lambda>\<xi>. { \<xi>\<lparr> dip := dip \<rparr> | dip. dip \<in> qD(store \<xi>) \<inter> vD(rt \<xi>) }\<rangle>
\<lbrakk>\<xi>. \<xi> \<lparr> data := hd(\<sigma>\<^bsub>queue\<^esub>(store \<xi>, dip \<xi>)) \<rparr>\<rbrakk>
unicast(\<lambda>\<xi>. the (nhop (rt \<xi>) (dip \<xi>)), \<lambda>\<xi>. pkt(data \<xi>, dip \<xi>, ip \<xi>)).
\<lbrakk>\<xi>. \<xi> \<lparr> store := the (drop (dip \<xi>) (store \<xi>)) \<rparr>\<rbrakk>
AODV()
\<triangleright> \<lbrakk>\<xi>. \<xi> \<lparr> dests := (\<lambda>rip. if (rip \<in> vD (rt \<xi>) \<and> nhop (rt \<xi>) rip = nhop (rt \<xi>) (dip \<xi>))
then Some (inc (sqn (rt \<xi>) rip)) else None) \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> rt := invalidate (rt \<xi>) (dests \<xi>) \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> store := setRRF (store \<xi>) (dests \<xi>)\<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> pre := \<Union>{ the (precs (rt \<xi>) rip) | rip. rip \<in> dom (dests \<xi>) } \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> dests := (\<lambda>rip. if ((dests \<xi>) rip \<noteq> None \<and> the (precs (rt \<xi>) rip) \<noteq> {})
then (dests \<xi>) rip else None) \<rparr>\<rbrakk>
groupcast(\<lambda>\<xi>. pre \<xi>, \<lambda>\<xi>. rerr(dests \<xi>, ip \<xi>)). AODV()
\<oplus> \<langle>\<lambda>\<xi>. { \<xi>\<lparr> dip := dip \<rparr>
| dip. dip \<in> qD(store \<xi>) - vD(rt \<xi>) \<and> the (\<sigma>\<^bsub>p-flag\<^esub>(store \<xi>, dip)) = req }\<rangle>
\<lbrakk>\<xi>. \<xi> \<lparr> store := unsetRRF (store \<xi>) (dip \<xi>) \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> sn := inc (sn \<xi>) \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> rreqid := nrreqid (rreqs \<xi>) (ip \<xi>) \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> rreqs := rreqs \<xi> \<union> {(ip \<xi>, rreqid \<xi>)} \<rparr>\<rbrakk>
broadcast(\<lambda>\<xi>. rreq(0, rreqid \<xi>, dip \<xi>, sqn (rt \<xi>) (dip \<xi>), sqnf (rt \<xi>) (dip \<xi>),
ip \<xi>, sn \<xi>, ip \<xi>)). AODV())"
| "\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PNewPkt = labelled PNewPkt (
\<langle>\<xi>. dip \<xi> = ip \<xi>\<rangle>
deliver(\<lambda>\<xi>. data \<xi>).AODV()
\<oplus> \<langle>\<xi>. dip \<xi> \<noteq> ip \<xi>\<rangle>
\<lbrakk>\<xi>. \<xi> \<lparr> store := add (data \<xi>) (dip \<xi>) (store \<xi>) \<rparr>\<rbrakk>
AODV())"
| "\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PPkt = labelled PPkt (
\<langle>\<xi>. dip \<xi> = ip \<xi>\<rangle>
deliver(\<lambda>\<xi>. data \<xi>).AODV()
\<oplus> \<langle>\<xi>. dip \<xi> \<noteq> ip \<xi>\<rangle>
(
\<langle>\<xi>. dip \<xi> \<in> vD (rt \<xi>)\<rangle>
unicast(\<lambda>\<xi>. the (nhop (rt \<xi>) (dip \<xi>)), \<lambda>\<xi>. pkt(data \<xi>, dip \<xi>, oip \<xi>)).AODV()
\<triangleright>
\<lbrakk>\<xi>. \<xi> \<lparr> dests := (\<lambda>rip. if (rip \<in> vD (rt \<xi>) \<and> nhop (rt \<xi>) rip = nhop (rt \<xi>) (dip \<xi>))
then Some (inc (sqn (rt \<xi>) rip)) else None) \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> rt := invalidate (rt \<xi>) (dests \<xi>) \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> store := setRRF (store \<xi>) (dests \<xi>)\<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> pre := \<Union>{ the (precs (rt \<xi>) rip) | rip. rip \<in> dom (dests \<xi>) } \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> dests := (\<lambda>rip. if ((dests \<xi>) rip \<noteq> None \<and> the (precs (rt \<xi>) rip) \<noteq> {})
then (dests \<xi>) rip else None) \<rparr>\<rbrakk>
groupcast(\<lambda>\<xi>. pre \<xi>, \<lambda>\<xi>. rerr(dests \<xi>, ip \<xi>)).AODV()
\<oplus> \<langle>\<xi>. dip \<xi> \<notin> vD (rt \<xi>)\<rangle>
(
\<langle>\<xi>. dip \<xi> \<in> iD (rt \<xi>)\<rangle>
groupcast(\<lambda>\<xi>. the (precs (rt \<xi>) (dip \<xi>)),
\<lambda>\<xi>. rerr([dip \<xi> \<mapsto> sqn (rt \<xi>) (dip \<xi>)], ip \<xi>)). AODV()
\<oplus> \<langle>\<xi>. dip \<xi> \<notin> iD (rt \<xi>)\<rangle>
AODV()
)
))"
| "\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PRreq = labelled PRreq (
\<langle>\<xi>. (oip \<xi>, rreqid \<xi>) \<in> rreqs \<xi>\<rangle>
AODV()
\<oplus> \<langle>\<xi>. (oip \<xi>, rreqid \<xi>) \<notin> rreqs \<xi>\<rangle>
\<lbrakk>\<xi>. \<xi> \<lparr> rt := update (rt \<xi>) (oip \<xi>) (osn \<xi>, kno, val, hops \<xi> + 1, sip \<xi>, {}) \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> rreqs := rreqs \<xi> \<union> {(oip \<xi>, rreqid \<xi>)} \<rparr>\<rbrakk>
(
\<langle>\<xi>. dip \<xi> = ip \<xi>\<rangle>
\<lbrakk>\<xi>. \<xi> \<lparr> sn := max (sn \<xi>) (dsn \<xi>) \<rparr>\<rbrakk>
unicast(\<lambda>\<xi>. the (nhop (rt \<xi>) (oip \<xi>)), \<lambda>\<xi>. rrep(0, dip \<xi>, sn \<xi>, oip \<xi>, ip \<xi>)).AODV()
\<triangleright>
\<lbrakk>\<xi>. \<xi> \<lparr> dests := (\<lambda>rip. if (rip \<in> vD (rt \<xi>) \<and> nhop (rt \<xi>) rip = nhop (rt \<xi>) (oip \<xi>))
then Some (inc (sqn (rt \<xi>) rip)) else None) \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> rt := invalidate (rt \<xi>) (dests \<xi>) \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> store := setRRF (store \<xi>) (dests \<xi>)\<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> pre := \<Union>{ the (precs (rt \<xi>) rip) | rip. rip \<in> dom (dests \<xi>) } \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> dests := (\<lambda>rip. if ((dests \<xi>) rip \<noteq> None \<and> the (precs (rt \<xi>) rip) \<noteq> {})
then (dests \<xi>) rip else None) \<rparr>\<rbrakk>
groupcast(\<lambda>\<xi>. pre \<xi>, \<lambda>\<xi>. rerr(dests \<xi>, ip \<xi>)).AODV()
\<oplus> \<langle>\<xi>. dip \<xi> \<noteq> ip \<xi>\<rangle>
(
\<langle>\<xi>. dip \<xi> \<in> vD (rt \<xi>) \<and> dsn \<xi> \<le> sqn (rt \<xi>) (dip \<xi>) \<and> sqnf (rt \<xi>) (dip \<xi>) = kno\<rangle>
\<lbrakk>\<xi>. \<xi> \<lparr> rt := the (addpreRT (rt \<xi>) (dip \<xi>) {sip \<xi>}) \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> rt := the (addpreRT (rt \<xi>) (oip \<xi>) {the (nhop (rt \<xi>) (dip \<xi>))}) \<rparr>\<rbrakk>
unicast(\<lambda>\<xi>. the (nhop (rt \<xi>) (oip \<xi>)), \<lambda>\<xi>. rrep(the (dhops (rt \<xi>) (dip \<xi>)), dip \<xi>,
sqn (rt \<xi>) (dip \<xi>), oip \<xi>, ip \<xi>)).
AODV()
\<triangleright>
\<lbrakk>\<xi>. \<xi> \<lparr> dests := (\<lambda>rip. if (rip \<in> vD (rt \<xi>) \<and> nhop (rt \<xi>) rip = nhop (rt \<xi>) (oip \<xi>))
then Some (inc (sqn (rt \<xi>) rip)) else None) \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> rt := invalidate (rt \<xi>) (dests \<xi>) \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> store := setRRF (store \<xi>) (dests \<xi>)\<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> pre := \<Union>{ the (precs (rt \<xi>) rip) | rip. rip \<in> dom (dests \<xi>) } \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> dests := (\<lambda>rip. if ((dests \<xi>) rip \<noteq> None \<and> the (precs (rt \<xi>) rip) \<noteq> {})
then (dests \<xi>) rip else None) \<rparr>\<rbrakk>
groupcast(\<lambda>\<xi>. pre \<xi>, \<lambda>\<xi>. rerr(dests \<xi>, ip \<xi>)).AODV()
\<oplus> \<langle>\<xi>. dip \<xi> \<notin> vD (rt \<xi>) \<or> sqn (rt \<xi>) (dip \<xi>) < dsn \<xi> \<or> sqnf (rt \<xi>) (dip \<xi>) = unk\<rangle>
broadcast(\<lambda>\<xi>. rreq(hops \<xi> + 1, rreqid \<xi>, dip \<xi>, max (sqn (rt \<xi>) (dip \<xi>)) (dsn \<xi>),
dsk \<xi>, oip \<xi>, osn \<xi>, ip \<xi>)).
AODV()
)
))"
| "\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PRrep = labelled PRrep (
\<langle>\<xi>. rt \<xi> \<noteq> update (rt \<xi>) (dip \<xi>) (dsn \<xi>, kno, val, hops \<xi> + 1, sip \<xi>, {}) \<rangle>
(
\<lbrakk>\<xi>. \<xi> \<lparr> rt := update (rt \<xi>) (dip \<xi>) (dsn \<xi>, kno, val, hops \<xi> + 1, sip \<xi>, {}) \<rparr> \<rbrakk>
(
\<langle>\<xi>. oip \<xi> = ip \<xi> \<rangle>
AODV()
\<oplus> \<langle>\<xi>. oip \<xi> \<noteq> ip \<xi> \<rangle>
(
\<langle>\<xi>. oip \<xi> \<in> vD (rt \<xi>)\<rangle>
\<lbrakk>\<xi>. \<xi> \<lparr> rt := the (addpreRT (rt \<xi>) (dip \<xi>) {the (nhop (rt \<xi>) (oip \<xi>))}) \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> rt := the (addpreRT (rt \<xi>) (the (nhop (rt \<xi>) (dip \<xi>)))
{the (nhop (rt \<xi>) (oip \<xi>))}) \<rparr>\<rbrakk>
unicast(\<lambda>\<xi>. the (nhop (rt \<xi>) (oip \<xi>)), \<lambda>\<xi>. rrep(hops \<xi> + 1, dip \<xi>, dsn \<xi>, oip \<xi>, ip \<xi>)).
AODV()
\<triangleright>
\<lbrakk>\<xi>. \<xi> \<lparr> dests := (\<lambda>rip. if (rip \<in> vD (rt \<xi>) \<and> nhop (rt \<xi>) rip = nhop (rt \<xi>) (oip \<xi>))
then Some (inc (sqn (rt \<xi>) rip)) else None) \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> rt := invalidate (rt \<xi>) (dests \<xi>) \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> store := setRRF (store \<xi>) (dests \<xi>)\<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> pre := \<Union>{ the (precs (rt \<xi>) rip) | rip. rip \<in> dom (dests \<xi>) } \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> dests := (\<lambda>rip. if ((dests \<xi>) rip \<noteq> None \<and> the (precs (rt \<xi>) rip) \<noteq> {})
then (dests \<xi>) rip else None) \<rparr>\<rbrakk>
groupcast(\<lambda>\<xi>. pre \<xi>, \<lambda>\<xi>. rerr(dests \<xi>, ip \<xi>)).AODV()
\<oplus> \<langle>\<xi>. oip \<xi> \<notin> vD (rt \<xi>)\<rangle>
AODV()
)
)
)
\<oplus> \<langle>\<xi>. rt \<xi> = update (rt \<xi>) (dip \<xi>) (dsn \<xi>, kno, val, hops \<xi> + 1, sip \<xi>, {}) \<rangle>
AODV()
)"
| "\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PRerr = labelled PRerr (
\<lbrakk>\<xi>. \<xi> \<lparr> dests := (\<lambda>rip. case (dests \<xi>) rip of None \<Rightarrow> None
| Some rsn \<Rightarrow> if rip \<in> vD (rt \<xi>) \<and> the (nhop (rt \<xi>) rip) = sip \<xi>
\<and> sqn (rt \<xi>) rip < rsn then Some rsn else None) \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> rt := invalidate (rt \<xi>) (dests \<xi>) \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> store := setRRF (store \<xi>) (dests \<xi>)\<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> pre := \<Union>{ the (precs (rt \<xi>) rip) | rip. rip \<in> dom (dests \<xi>) } \<rparr>\<rbrakk>
\<lbrakk>\<xi>. \<xi> \<lparr> dests := (\<lambda>rip. if ((dests \<xi>) rip \<noteq> None \<and> the (precs (rt \<xi>) rip) \<noteq> {})
then (dests \<xi>) rip else None) \<rparr>\<rbrakk>
groupcast(\<lambda>\<xi>. pre \<xi>, \<lambda>\<xi>. rerr(dests \<xi>, ip \<xi>)). AODV())"
declare \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V.simps [simp del, code del]
lemmas \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_simps [simp, code] = \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V.simps [simplified]
fun \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_skeleton
where
"\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_skeleton PAodv = seqp_skeleton (\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PAodv)"
| "\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_skeleton PNewPkt = seqp_skeleton (\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PNewPkt)"
| "\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_skeleton PPkt = seqp_skeleton (\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PPkt)"
| "\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_skeleton PRreq = seqp_skeleton (\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PRreq)"
| "\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_skeleton PRrep = seqp_skeleton (\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PRrep)"
| "\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_skeleton PRerr = seqp_skeleton (\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PRerr)"
lemma \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_skeleton_wf [simp]:
"wellformed \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_skeleton"
proof (rule, intro allI)
fix pn pn'
show "call(pn') \<notin> stermsl (\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_skeleton pn)"
by (cases pn) simp_all
qed
declare \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_skeleton.simps [simp del, code del]
lemmas \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_skeleton_simps [simp, code]
= \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_skeleton.simps [simplified \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_simps seqp_skeleton.simps]
lemma aodv_proc_cases [dest]:
fixes p pn
shows "p \<in> ctermsl (\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V pn) \<Longrightarrow>
(p \<in> ctermsl (\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PAodv) \<or>
p \<in> ctermsl (\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PNewPkt) \<or>
p \<in> ctermsl (\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PPkt) \<or>
p \<in> ctermsl (\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PRreq) \<or>
p \<in> ctermsl (\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PRrep) \<or>
p \<in> ctermsl (\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PRerr))"
by (cases pn) simp_all
definition \<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V :: "ip \<Rightarrow> (state \<times> (state, msg, pseqp, pseqp label) seqp) set"
where "\<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V i \<equiv> {(aodv_init i, \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PAodv)}"
abbreviation paodv
:: "ip \<Rightarrow> (state \<times> (state, msg, pseqp, pseqp label) seqp, msg seq_action) automaton"
where
"paodv i \<equiv> \<lparr> init = \<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V i, trans = seqp_sos \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V \<rparr>"
lemma aodv_trans: "trans (paodv i) = seqp_sos \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V"
by simp
lemma aodv_control_within [simp]: "control_within \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (init (paodv i))"
unfolding \<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_def by (rule control_withinI) (auto simp del: \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_simps)
lemma aodv_wf [simp]:
"wellformed \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V"
proof (rule, intro allI)
fix pn pn'
show "call(pn') \<notin> stermsl (\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V pn)"
by (cases pn) simp_all
qed
lemmas aodv_labels_not_empty [simp] = labels_not_empty [OF aodv_wf]
lemma aodv_ex_label [intro]: "\<exists>l. l\<in>labels \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V p"
by (metis aodv_labels_not_empty all_not_in_conv)
lemma aodv_ex_labelE [elim]:
assumes "\<forall>l\<in>labels \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V p. P l p"
and "\<exists>p l. P l p \<Longrightarrow> Q"
shows "Q"
using assms by (metis aodv_ex_label)
lemma aodv_simple_labels [simp]: "simple_labels \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V"
proof
fix pn p
assume "p\<in>subterms(\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V pn)"
thus "\<exists>!l. labels \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V p = {l}"
by (cases pn) (simp_all cong: seqp_congs | elim disjE)+
qed
lemma \<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_labels [simp]: "(\<xi>, p) \<in> \<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V i \<Longrightarrow> labels \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V p = {PAodv-:0}"
unfolding \<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_def by simp
lemma aodv_init_kD_empty [simp]:
"(\<xi>, p) \<in> \<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V i \<Longrightarrow> kD (rt \<xi>) = {}"
unfolding \<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_def kD_def by simp
lemma aodv_init_sip_not_ip [simp]: "\<not>(sip (aodv_init i) = i)" by simp
lemma aodv_init_sip_not_ip' [simp]:
assumes "(\<xi>, p) \<in> \<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V i"
shows "sip \<xi> \<noteq> ip \<xi>"
using assms unfolding \<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_def by simp
lemma aodv_init_sip_not_i [simp]:
assumes "(\<xi>, p) \<in> \<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V i"
shows "sip \<xi> \<noteq> i"
using assms unfolding \<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_def by simp
lemma clear_locals_sip_not_ip':
assumes "ip \<xi> = i"
shows "\<not>(sip (clear_locals \<xi>) = i)"
using assms by auto
text \<open>Stop the simplifier from descending into process terms.\<close>
declare seqp_congs [cong]
text \<open>Configure the main invariant tactic for AODV.\<close>
declare
\<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_simps [cterms_env]
aodv_proc_cases [ctermsl_cases]
seq_invariant_ctermsI [OF aodv_wf aodv_control_within aodv_simple_labels aodv_trans,
cterms_intros]
seq_step_invariant_ctermsI [OF aodv_wf aodv_control_within aodv_simple_labels aodv_trans,
cterms_intros]
end
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