IOA/example/Impl.thy

added IOA to isabelle/HOL

Impl = Sender + Receiver + Channels +

types 

'm impl_state 
= "'m sender_state * 'm receiver_state * 'm packet multiset * bool multiset"
(*  sender_state   *  receiver_state   *    srch_state      * rsch_state *)


consts
 impl_ioa    :: "('m action, 'm impl_state)ioa"
 sen         :: "'m impl_state => 'm sender_state"
 rec         :: "'m impl_state => 'm receiver_state"
 srch        :: "'m impl_state => 'm packet multiset"
 rsch        :: "'m impl_state => bool multiset"
 inv1, inv2, 
 inv3, inv4  :: "'m impl_state => bool"
 hdr_sum     :: "'m packet multiset => bool => nat"

rules

 impl_def
  "impl_ioa == (sender_ioa || receiver_ioa || srch_ioa || rsch_ioa)"

 sen_def   "sen == fst"
 rec_def   "rec == fst o snd"
 srch_def "srch == fst o snd o snd"
 rsch_def "rsch == snd o snd o snd"

hdr_sum_def
   "hdr_sum(M,b) == countm(M,%pkt.hdr(pkt) = b)"

(* Lemma 5.1 *)
inv1_def 
  "inv1(s) ==                                                                 \
 \   (!b. count(rsent(rec(s)),b) = count(srcvd(sen(s)),b) + count(rsch(s),b)) \
 \ & (!b. count(ssent(sen(s)),b)                                              \
 \        = hdr_sum(rrcvd(rec(s)),b) + hdr_sum(srch(s),b))"

(* Lemma 5.2 *)
 inv2_def "inv2(s) ==                                                   \
\  (rbit(rec(s)) = sbit(sen(s)) &                                       \
\   ssending(sen(s)) &                                                  \
\   count(rsent(rec(s)),~sbit(sen(s))) <= count(ssent(sen(s)),~sbit(sen(s))) &\
\   count(ssent(sen(s)),~sbit(sen(s))) <= count(rsent(rec(s)),sbit(sen(s))))  \
\   |                                                                   \
\  (rbit(rec(s)) = (~sbit(sen(s))) &                                    \
\   rsending(rec(s)) &                                                  \
\   count(ssent(sen(s)),~sbit(sen(s))) <= count(rsent(rec(s)),sbit(sen(s))) &       \
\   count(rsent(rec(s)),sbit(sen(s))) <= count(ssent(sen(s)),sbit(sen(s))))"

(* Lemma 5.3 *)
 inv3_def "inv3(s) ==                                                   \
\   rbit(rec(s)) = sbit(sen(s))                                         \
\   --> (!m. sq(sen(s))=[] | m ~= hd(sq(sen(s)))                        \
\        -->  count(rrcvd(rec(s)),<sbit(sen(s)),m>)                     \
\             + count(srch(s),<sbit(sen(s)),m>)                         \
\            <= count(rsent(rec(s)),~sbit(sen(s))))"

(* Lemma 5.4 *)
 inv4_def "inv4(s) == rbit(rec(s)) = (~sbit(sen(s))) --> sq(sen(s)) ~= []"

end