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1 Impl = Sender + Receiver + Channels + |
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2 |
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3 types |
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4 |
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5 'm impl_state |
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6 = "'m sender_state * 'm receiver_state * 'm packet multiset * bool multiset" |
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7 (* sender_state * receiver_state * srch_state * rsch_state *) |
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8 |
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9 |
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10 consts |
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11 impl_ioa :: "('m action, 'm impl_state)ioa" |
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12 sen :: "'m impl_state => 'm sender_state" |
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13 rec :: "'m impl_state => 'm receiver_state" |
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14 srch :: "'m impl_state => 'm packet multiset" |
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15 rsch :: "'m impl_state => bool multiset" |
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16 inv1, inv2, |
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17 inv3, inv4 :: "'m impl_state => bool" |
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18 hdr_sum :: "'m packet multiset => bool => nat" |
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19 |
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20 rules |
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21 |
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22 impl_def |
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23 "impl_ioa == (sender_ioa || receiver_ioa || srch_ioa || rsch_ioa)" |
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24 |
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25 sen_def "sen == fst" |
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26 rec_def "rec == fst o snd" |
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27 srch_def "srch == fst o snd o snd" |
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28 rsch_def "rsch == snd o snd o snd" |
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29 |
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30 hdr_sum_def |
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31 "hdr_sum(M,b) == countm(M,%pkt.hdr(pkt) = b)" |
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32 |
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33 (* Lemma 5.1 *) |
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34 inv1_def |
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35 "inv1(s) == \ |
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36 \ (!b. count(rsent(rec(s)),b) = count(srcvd(sen(s)),b) + count(rsch(s),b)) \ |
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37 \ & (!b. count(ssent(sen(s)),b) \ |
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38 \ = hdr_sum(rrcvd(rec(s)),b) + hdr_sum(srch(s),b))" |
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39 |
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40 (* Lemma 5.2 *) |
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41 inv2_def "inv2(s) == \ |
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42 \ (rbit(rec(s)) = sbit(sen(s)) & \ |
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43 \ ssending(sen(s)) & \ |
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44 \ count(rsent(rec(s)),~sbit(sen(s))) <= count(ssent(sen(s)),~sbit(sen(s))) &\ |
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45 \ count(ssent(sen(s)),~sbit(sen(s))) <= count(rsent(rec(s)),sbit(sen(s)))) \ |
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46 \ | \ |
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47 \ (rbit(rec(s)) = (~sbit(sen(s))) & \ |
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48 \ rsending(rec(s)) & \ |
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49 \ count(ssent(sen(s)),~sbit(sen(s))) <= count(rsent(rec(s)),sbit(sen(s))) & \ |
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50 \ count(rsent(rec(s)),sbit(sen(s))) <= count(ssent(sen(s)),sbit(sen(s))))" |
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51 |
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52 (* Lemma 5.3 *) |
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53 inv3_def "inv3(s) == \ |
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54 \ rbit(rec(s)) = sbit(sen(s)) \ |
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55 \ --> (!m. sq(sen(s))=[] | m ~= hd(sq(sen(s))) \ |
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56 \ --> count(rrcvd(rec(s)),<sbit(sen(s)),m>) \ |
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57 \ + count(srch(s),<sbit(sen(s)),m>) \ |
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58 \ <= count(rsent(rec(s)),~sbit(sen(s))))" |
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59 |
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60 (* Lemma 5.4 *) |
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61 inv4_def "inv4(s) == rbit(rec(s)) = (~sbit(sen(s))) --> sq(sen(s)) ~= []" |
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62 |
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63 end |