Step
*
1
1
of Lemma
pv11_p1_A5_C2
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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs_uid : Id ─→ ℤ@i
16. reps : bag(Id)@i
17. b : pv11_p1_Ballot_Num()@i
18. s : ℤ@i
19. p : Cmd@i
20. p' : Cmd@i
21. as : bag(Id)@i
22. cs : bag(Id)@i
23. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
24. Inj(Id;ℤ;ldrs_uid)@i
25. accpts = (as + cs) ∈ bag(Id)@i
26. #(as) < pv11_p1_threshold(accpts)@i
27. ∀a:Id
(a ↓∈ cs ⇒ (↓∃e':E. ((a = loc(e') ∈ Id) ∧ (<b, s, p> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))))))@i
28. e : E@i
29. ∀e1:E
((e1 < e)
⇒ (∀b':pv11_p1_Ballot_Num()
((↑(b < b'))
⇒ (∀l,i:Id.
(((<b', s, p'> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e1)))
∨ pv11_p1_p2a'send(Cmd;f) i <l, b', s, p'> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e1))
⇒ (p = p' ∈ Cmd))))))
30. b' : pv11_p1_Ballot_Num()@i
31. ↑(b < b')@i
32. l : Id@i
33. i : Id@i
34. (<b', s, p'> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)))@i
⊢ p = p' ∈ Cmd
BY
{ ((InstLemma `pv11_p1_acc_p2a` [⌈Cmd⌉;⌈f⌉;⌈es⌉;⌈e⌉;⌈ldrs_uid⌉;⌈<fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e))
, snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e))
>⌉;⌈b'⌉;⌈s⌉;⌈p'⌉]⋅
THENA (Auto THEN BLemma `pv11_p1_AcceptorState-classrel` THEN Auto THEN RWO "pair-eta<" 0 THEN Auto)
)
THEN Reduce (-1)
THEN (D (-1) THENA Auto)
THEN SquashExRepD
THEN (UseMessageConstraint' (-1) THENA Auto)
THEN InstHyp [⌈e'@0⌉;⌈b'⌉;⌈l1⌉;⌈loc(e')⌉] (-14)⋅
THEN Auto) }
1
.....antecedent.....
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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs_uid : Id ─→ ℤ@i
16. reps : bag(Id)@i
17. b : pv11_p1_Ballot_Num()@i
18. s : ℤ@i
19. p : Cmd@i
20. p' : Cmd@i
21. as : bag(Id)@i
22. cs : bag(Id)@i
23. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
24. Inj(Id;ℤ;ldrs_uid)@i
25. accpts = (as + cs) ∈ bag(Id)@i
26. #(as) < pv11_p1_threshold(accpts)@i
27. ∀a:Id
(a ↓∈ cs ⇒ (↓∃e':E. ((a = loc(e') ∈ Id) ∧ (<b, s, p> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))))))@i
28. e : E@i
29. ∀e1:E
((e1 < e)
⇒ (∀b':pv11_p1_Ballot_Num()
((↑(b < b'))
⇒ (∀l,i:Id.
(((<b', s, p'> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e1)))
∨ pv11_p1_p2a'send(Cmd;f) i <l, b', s, p'> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e1))
⇒ (p = p' ∈ Cmd))))))
30. b' : pv11_p1_Ballot_Num()@i
31. ↑(b < b')@i
32. l : Id@i
33. i : Id@i
34. (<b', s, p'> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)))@i
35. e' : E
36. l1 : Id
37. e' ≤loc e
38. <l1, b', s, p'> ∈ pv11_p1_p2a'base(Cmd;f)(e')
39. e'@0 : E
40. delay : ℤ
41. (e'@0 < e')
42. make-msg-interface(delay;loc(e');mk-msg(msg-authentic(info(e'));``pv11_p1 p2a``;<l1, b', s, p'>)) ∈ pv11_p1_main(Cmd\000C;accpts;ldrs;ldrs_uid;reps;f)(e'@0)
⊢ (<b', s, p'> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'@0)))
∨ pv11_p1_p2a'send(Cmd;f) loc(e') <l1, b', s, p'> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e'@0)
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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i
16. reps : bag(Id)@i
17. b : pv11\_p1\_Ballot\_Num()@i
18. s : \mBbbZ{}@i
19. p : Cmd@i
20. p' : Cmd@i
21. as : bag(Id)@i
22. cs : bag(Id)@i
23. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i
24. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i
25. accpts = (as + cs)@i
26. \#(as) < pv11\_p1\_threshold(accpts)@i
27. \mforall{}a:Id
(a \mdownarrow{}\mmember{} cs
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E
((a = loc(e')) \mwedge{} (<b, s, p> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid\mbackslash{}ff\000C7d$;f;es;e'))))))@i
28. e : E@i
29. \mforall{}e1:E
((e1 < e)
{}\mRightarrow{} (\mforall{}b':pv11\_p1\_Ballot\_Num()
((\muparrow{}(b < b'))
{}\mRightarrow{} (\mforall{}l,i:Id.
(((<b', s, p'> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$\000C;f;es;e1)))
\mvee{} pv11\_p1\_p2a'send(Cmd;f) i <l, b', s, p'> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs\mbackslash{}ff2\000C4_{uid}$;reps;f)(e1))
{}\mRightarrow{} (p = p'))))))
30. b' : pv11\_p1\_Ballot\_Num()@i
31. \muparrow{}(b < b')@i
32. l : Id@i
33. i : Id@i
34. (<b', s, p'> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e)))@i
\mvdash{} p = p'
By
Latex:
((InstLemma `pv11\_p1\_acc\_p2a` [\mkleeneopen{}Cmd\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}ldrs$_{uid}$\mkleeneclose{};
\mkleeneopen{}<fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e))
, snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e))
>\mkleeneclose{};\mkleeneopen{}b'\mkleeneclose{};\mkleeneopen{}s\mkleeneclose{};\mkleeneopen{}p'\mkleeneclose{}]\mcdot{}
THENA (Auto
THEN BLemma `pv11\_p1\_AcceptorState-classrel`
THEN Auto
THEN RWO "pair-eta<" 0
THEN Auto)
)
THEN Reduce (-1)
THEN (D (-1) THENA Auto)
THEN SquashExRepD
THEN (UseMessageConstraint' (-1) THENA Auto)
THEN InstHyp [\mkleeneopen{}e'@0\mkleeneclose{};\mkleeneopen{}b'\mkleeneclose{};\mkleeneopen{}l1\mkleeneclose{};\mkleeneopen{}loc(e')\mkleeneclose{}] (-14)\mcdot{}
THEN Auto)
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