Step
*
1
4
of Lemma
pv11_p1_acc_rcv_p2a
1. Cmd : {T:Type| valueall-type(T)} @i'
2. f : pv11_p1_headers_type{i:l}(Cmd)@i'
3. (f [decision]) = (ℤ × Cmd) ∈ Type
4. (f [propose]) = (ℤ × Cmd) ∈ Type
5. (f ``pv11_p1 adopted``) = (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)) ∈ Type
6. (f ``pv11_p1 preempted``) = pv11_p1_Ballot_Num() ∈ Type
7. (f ``pv11_p1 p2b``) = (Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num()) ∈ Type
8. (f ``pv11_p1 p2a``) = (Id × pv11_p1_Ballot_Num() × ℤ × Cmd) ∈ Type
9. (f ``pv11_p1 p1b``)
= (Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
∈ Type
10. (f ``pv11_p1 p1a``) = (Id × pv11_p1_Ballot_Num()) ∈ Type
11. f ∈ Name ─→ Type
12. es : EO+(Message(f))@i'
13. e : E@i
14. ¬↑first(e)
15. ¬↑e ∈b pv11_p1_p1a'base(Cmd;f)
16. accpts : bag(Id)@i
17. ldrs : bag(Id)@i
18. ldrs_uid : Id ─→ ℤ@i
19. reps : bag(Id)@i
20. b : pv11_p1_Ballot_Num()@i
21. s : ℤ@i
22. c : Cmd@i
23. Inj(Id;ℤ;ldrs_uid)@i
24. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
25. header(e) = ``pv11_p1 p2a`` ∈ Name@i
26. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)@i
27. ∃e':E. ∃i,l:Id. pv11_p1_p2a'send(Cmd;f) i <l, snd(msgval(e))> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e')@i
28. ↑e ∈b pv11_p1_p2a'base(Cmd;f)
29. ↑e ∈b pv11_p1_p2a'base(Cmd;f)
⊢ ((fst((pv11_p1_on_p2a(Cmd;ldrs_uid) loc(e) pv11_p1_p2a'base(Cmd;f)@e
pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))))
= (fst(snd(msgval(e))))
∈ pv11_p1_Ballot_Num())
⇒ (↓(snd(msgval(e)) ∈ snd((pv11_p1_on_p2a(Cmd;ldrs_uid) loc(e) pv11_p1_p2a'base(Cmd;f)@e
pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))))))
BY
{ (RepUR ``pv11_p1_on_p2a let pv11_p1_init_accepted pv11_p1_p2a'base`` 0
THEN RWO "classfun-res-base" 0
THEN Try (CpltAuto)
THEN Repeat (AutoPairEta [1;2;1;1] 0)
THEN AutoSplit
THEN (D 0 THENA Auto)) }
1
1. Cmd : {T:Type| valueall-type(T)} @i'
2. f : pv11_p1_headers_type{i:l}(Cmd)@i'
3. (f [decision]) = (ℤ × Cmd) ∈ Type
4. (f [propose]) = (ℤ × Cmd) ∈ Type
5. (f ``pv11_p1 adopted``) = (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)) ∈ Type
6. (f ``pv11_p1 preempted``) = pv11_p1_Ballot_Num() ∈ Type
7. (f ``pv11_p1 p2b``) = (Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num()) ∈ Type
8. (f ``pv11_p1 p2a``) = (Id × pv11_p1_Ballot_Num() × ℤ × Cmd) ∈ Type
9. (f ``pv11_p1 p1b``)
= (Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
∈ Type
10. (f ``pv11_p1 p1a``) = (Id × pv11_p1_Ballot_Num()) ∈ Type
11. f ∈ Name ─→ Type
12. es : EO+(Message(f))@i'
13. e : E@i
14. ¬↑first(e)
15. ¬↑e ∈b pv11_p1_p1a'base(Cmd;f)
16. accpts : bag(Id)@i
17. ldrs : bag(Id)@i
18. ldrs_uid : Id ─→ ℤ@i
19. reps : bag(Id)@i
20. b : pv11_p1_Ballot_Num()@i
21. s : ℤ@i
22. c : Cmd@i
23. Inj(Id;ℤ;ldrs_uid)@i
24. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
25. header(e) = ``pv11_p1 p2a`` ∈ Name@i
26. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)@i
27. ∃e':E. ∃i,l:Id. pv11_p1_p2a'send(Cmd;f) i <l, snd(msgval(e))> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e')@i
28. ↑e ∈b pv11_p1_p2a'base(Cmd;f)
29. ↑e ∈b pv11_p1_p2a'base(Cmd;f)
30. ↑(pv11_p1_leq_bnum(ldrs_uid) (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))) (fst(snd(msgval(e)))))
31. (pv11_p1_max_bnum(ldrs_uid) (fst(snd(msgval(e)))) (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))))
= (fst(snd(msgval(e))))
∈ pv11_p1_Ballot_Num()@i
⊢ ↓(<fst(snd(msgval(e))), snd(snd(msgval(e)))> ∈ pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd)
<fst(snd(msgval(e))), snd(snd(msgval(e)))>
(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))))
2
1. Cmd : {T:Type| valueall-type(T)} @i'
2. f : pv11_p1_headers_type{i:l}(Cmd)@i'
3. (f [decision]) = (ℤ × Cmd) ∈ Type
4. (f [propose]) = (ℤ × Cmd) ∈ Type
5. (f ``pv11_p1 adopted``) = (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)) ∈ Type
6. (f ``pv11_p1 preempted``) = pv11_p1_Ballot_Num() ∈ Type
7. (f ``pv11_p1 p2b``) = (Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num()) ∈ Type
8. (f ``pv11_p1 p2a``) = (Id × pv11_p1_Ballot_Num() × ℤ × Cmd) ∈ Type
9. (f ``pv11_p1 p1b``)
= (Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
∈ Type
10. (f ``pv11_p1 p1a``) = (Id × pv11_p1_Ballot_Num()) ∈ Type
11. f ∈ Name ─→ Type
12. es : EO+(Message(f))@i'
13. e : E@i
14. ¬↑first(e)
15. ¬↑e ∈b pv11_p1_p1a'base(Cmd;f)
16. accpts : bag(Id)@i
17. ldrs : bag(Id)@i
18. ldrs_uid : Id ─→ ℤ@i
19. ¬↑(pv11_p1_leq_bnum(ldrs_uid) (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))) (fst(snd(msgval(e)))))
20. reps : bag(Id)@i
21. b : pv11_p1_Ballot_Num()@i
22. s : ℤ@i
23. c : Cmd@i
24. Inj(Id;ℤ;ldrs_uid)@i
25. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
26. header(e) = ``pv11_p1 p2a`` ∈ Name@i
27. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)@i
28. ∃e':E. ∃i,l:Id. pv11_p1_p2a'send(Cmd;f) i <l, snd(msgval(e))> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e')@i
29. ↑e ∈b pv11_p1_p2a'base(Cmd;f)
30. ↑e ∈b pv11_p1_p2a'base(Cmd;f)
31. (pv11_p1_max_bnum(ldrs_uid) (fst(snd(msgval(e)))) (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))))
= (fst(snd(msgval(e))))
∈ pv11_p1_Ballot_Num()@i
⊢ ↓(<fst(snd(msgval(e))), snd(snd(msgval(e)))> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))))
Latex:
Latex:
1. Cmd : \{T:Type| valueall-type(T)\} @i'
2. f : pv11\_p1\_headers\_type\{i:l\}(Cmd)@i'
3. (f [decision]) = (\mBbbZ{} \mtimes{} Cmd)
4. (f [propose]) = (\mBbbZ{} \mtimes{} Cmd)
5. (f ``pv11\_p1 adopted``) = (pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))
6. (f ``pv11\_p1 preempted``) = pv11\_p1\_Ballot\_Num()
7. (f ``pv11\_p1 p2b``) = (Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num())
8. (f ``pv11\_p1 p2a``) = (Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd)
9. (f ``pv11\_p1 p1b``)
= (Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))
10. (f ``pv11\_p1 p1a``) = (Id \mtimes{} pv11\_p1\_Ballot\_Num())
11. f \mmember{} Name {}\mrightarrow{} Type
12. es : EO+(Message(f))@i'
13. e : E@i
14. \mneg{}\muparrow{}first(e)
15. \mneg{}\muparrow{}e \mmember{}\msubb{} pv11\_p1\_p1a'base(Cmd;f)
16. accpts : bag(Id)@i
17. ldrs : bag(Id)@i
18. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i
19. reps : bag(Id)@i
20. b : pv11\_p1\_Ballot\_Num()@i
21. s : \mBbbZ{}@i
22. c : Cmd@i
23. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i
24. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i
25. header(e) = ``pv11\_p1 p2a``@i
26. has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd)@i
27. \mexists{}e':E. \mexists{}i,l:Id. pv11\_p1\_p2a'send(Cmd;f) i <l, snd(msgval(e))> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldr\000Cs$_{uid}$;reps;f)(e')@i
28. \muparrow{}e \mmember{}\msubb{} pv11\_p1\_p2a'base(Cmd;f)
29. \muparrow{}e \mmember{}\msubb{} pv11\_p1\_p2a'base(Cmd;f)
\mvdash{} ((fst((pv11\_p1\_on\_p2a(Cmd;ldrs$_{uid}$) loc(e) pv11\_p1\_p2a'base(Cmd;f)@e
pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;pred(e)))))
= (fst(snd(msgval(e)))))
{}\mRightarrow{} (\mdownarrow{}(snd(msgval(e)) \mmember{} snd((pv11\_p1\_on\_p2a(Cmd;ldrs$_{uid}$) loc(e) pv11\_p1\_p2a'\000Cbase(Cmd;f)@e
pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;pred(\000Ce))))))
By
Latex:
(RepUR ``pv11\_p1\_on\_p2a let pv11\_p1\_init\_accepted pv11\_p1\_p2a'base`` 0
THEN RWO "classfun-res-base" 0
THEN Try (CpltAuto)
THEN Repeat (AutoPairEta [1;2;1;1] 0)
THEN AutoSplit
THEN (D 0 THENA Auto))
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