Step
*
1
4
1
of Lemma
pv11_p1_scout_state_from_p1bs
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. start : E@i
14. accpts : bag(Id)@i
15. ldrs : bag(Id)@i
16. ldrs_uid : Id ─→ ℤ@i
17. reps : bag(Id)@i
18. Inj(Id;ℤ;ldrs_uid)@i
19. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
20. e : E@i
21. ∀e1:E
((e1 < e)
⇒ start ≤loc e1
⇒ (∀bnum:pv11_p1_Ballot_Num(). ∀waitfor:bag(Id). ∀pvalues:(pv11_p1_Ballot_Num() × ℤ × Cmd) List.
((<waitfor, pvalues>
= pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;e1)
∈ (bag(Id) × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
⇒ (↓∃L:E List
((waitfor = [i∈accpts|¬bi ∈b map(λe.loc(e);L))] ∈ bag(Id))
∧ (∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd
((pv ∈ pvalues)
⇐⇒ (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L)))))
∧ (∀e':E
((e' ∈ L)
⇒ ((e' < e1)
∧ loc(e') ↓∈ accpts
∧ (bnum
= (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')))
∈ pv11_p1_Ballot_Num())))))))))
22. start ≤loc e @i
23. bnum : pv11_p1_Ballot_Num()@i
24. waitfor : bag(Id)@i
25. pvalues : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
26. <waitfor, pvalues>
= pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))
∈ (bag(Id) × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
27. ¬↑e ∈b pv11_p1_p1b'base(Cmd;f)
28. ¬↑first(e)
29. ¬↑first(e)
30. start ≤loc pred(e)
31. ↓∃L:E List
(((fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))))
= [i∈accpts|¬bi ∈b map(λe.loc(e);L))]
∈ bag(Id))
∧ (∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd
((pv ∈ snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))))
⇐⇒ (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L)))))
∧ (∀e':E
((e' ∈ L)
⇒ ((e' < pred(e))
∧ loc(e') ↓∈ accpts
∧ (bnum = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num())))))
32. pred(e) = pred(e) ∈ E
33. pred(e) = pred(e) ∈ E
34. pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))
= pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))
∈ (bag(Id) × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
⊢ ↓∃L:E List
((waitfor = [i∈accpts|¬bi ∈b map(λe.loc(e);L))] ∈ bag(Id))
∧ (∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd
((pv ∈ pvalues) ⇐⇒ (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L)))))
∧ (∀e':E
((e' ∈ L)
⇒ ((e' < e)
∧ loc(e') ↓∈ accpts
∧ (bnum = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num())))))
BY
{ (SimpEqPairs THEN RepeatFor 2 (ParallelOp -5) THEN 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. start : E@i
14. accpts : bag(Id)@i
15. ldrs : bag(Id)@i
16. ldrs_uid : Id ─→ ℤ@i
17. reps : bag(Id)@i
18. Inj(Id;ℤ;ldrs_uid)@i
19. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
20. e : E@i
21. ∀e1:E
((e1 < e)
⇒ start ≤loc e1
⇒ (∀bnum:pv11_p1_Ballot_Num(). ∀waitfor:bag(Id). ∀pvalues:(pv11_p1_Ballot_Num() × ℤ × Cmd) List.
((<waitfor, pvalues>
= pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;e1)
∈ (bag(Id) × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
⇒ (↓∃L:E List
((waitfor = [i∈accpts|¬bi ∈b map(λe.loc(e);L))] ∈ bag(Id))
∧ (∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd
((pv ∈ pvalues)
⇐⇒ (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L)))))
∧ (∀e':E
((e' ∈ L)
⇒ ((e' < e1)
∧ loc(e') ↓∈ accpts
∧ (bnum
= (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')))
∈ pv11_p1_Ballot_Num())))))))))
22. start ≤loc e @i
23. bnum : pv11_p1_Ballot_Num()@i
24. waitfor : bag(Id)@i
25. pvalues : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
26. waitfor = (fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e)))) ∈ bag(Id)
27. pvalues = (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e)))) ∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
28. ¬↑e ∈b pv11_p1_p1b'base(Cmd;f)
29. ¬↑first(e)
30. ¬↑first(e)
31. start ≤loc pred(e)
32. L : E List
33. (fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e)))) = [i∈accpts|¬bi ∈b map(λe.loc(e);L))] ∈ bag(Id)
34. ∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd
((pv ∈ snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))))
⇐⇒ (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L))))
35. ∀e':E
((e' ∈ L)
⇒ ((e' < pred(e))
∧ loc(e') ↓∈ accpts
∧ (bnum = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num())))
36. pred(e) = pred(e) ∈ E
37. pred(e) = pred(e) ∈ E
38. (fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))))
= (fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))))
∈ bag(Id)
39. (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))))
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
40. waitfor = [i∈accpts|¬bi ∈b map(λe.loc(e);L))] ∈ bag(Id)
41. ∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd
((pv ∈ pvalues) ⇐⇒ (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L))))
42. e' : E@i
43. (e' ∈ L)@i
⊢ (e' < 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. start : E@i
14. accpts : bag(Id)@i
15. ldrs : bag(Id)@i
16. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i
17. reps : bag(Id)@i
18. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i
19. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i
20. e : E@i
21. \mforall{}e1:E
((e1 < e)
{}\mRightarrow{} start \mleq{}loc e1
{}\mRightarrow{} (\mforall{}bnum:pv11\_p1\_Ballot\_Num(). \mforall{}waitfor:bag(Id).
\mforall{}pvalues:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List.
((<waitfor, pvalues> = pv11\_p1\_ScoutStateFun(Cmd;accpts;f;bnum;es.start;e1))
{}\mRightarrow{} (\mdownarrow{}\mexists{}L:E List
((waitfor = [i\mmember{}accpts|\mneg{}\msubb{}i \mmember{}\msubb{} map(\mlambda{}e.loc(e);L))])
\mwedge{} (\mforall{}pv:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd
((pv \mmember{} pvalues)
\mLeftarrow{}{}\mRightarrow{} (pv \mmember{} concat(map(\mlambda{}e.(snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{\000Cuid}$;f;es;e)));
L)))))
\mwedge{} (\mforall{}e':E
((e' \mmember{} L)
{}\mRightarrow{} ((e' < e1)
\mwedge{} loc(e') \mdownarrow{}\mmember{} accpts
\mwedge{} (bnum = (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}\mbackslash{}ff\000C24;f;es;e'))))))))))))
22. start \mleq{}loc e @i
23. bnum : pv11\_p1\_Ballot\_Num()@i
24. waitfor : bag(Id)@i
25. pvalues : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
26. <waitfor, pvalues> = pv11\_p1\_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))
27. \mneg{}\muparrow{}e \mmember{}\msubb{} pv11\_p1\_p1b'base(Cmd;f)
28. \mneg{}\muparrow{}first(e)
29. \mneg{}\muparrow{}first(e)
30. start \mleq{}loc pred(e)
31. \mdownarrow{}\mexists{}L:E List
(((fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))))
= [i\mmember{}accpts|\mneg{}\msubb{}i \mmember{}\msubb{} map(\mlambda{}e.loc(e);L))])
\mwedge{} (\mforall{}pv:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd
((pv \mmember{} snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))))
\mLeftarrow{}{}\mRightarrow{} (pv \mmember{} concat(map(\mlambda{}e.(snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}\mbackslash{}ff2\000C4;f;es;e)));L)))))
\mwedge{} (\mforall{}e':E
((e' \mmember{} L)
{}\mRightarrow{} ((e' < pred(e))
\mwedge{} loc(e') \mdownarrow{}\mmember{} accpts
\mwedge{} (bnum = (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e'))\000C))))))
32. pred(e) = pred(e)
33. pred(e) = pred(e)
34. pv11\_p1\_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))
= pv11\_p1\_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))
\mvdash{} \mdownarrow{}\mexists{}L:E List
((waitfor = [i\mmember{}accpts|\mneg{}\msubb{}i \mmember{}\msubb{} map(\mlambda{}e.loc(e);L))])
\mwedge{} (\mforall{}pv:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd
((pv \mmember{} pvalues)
\mLeftarrow{}{}\mRightarrow{} (pv \mmember{} concat(map(\mlambda{}e.(snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;\000Cf;es;e)));L)))))
\mwedge{} (\mforall{}e':E
((e' \mmember{} L)
{}\mRightarrow{} ((e' < e)
\mwedge{} loc(e') \mdownarrow{}\mmember{} accpts
\mwedge{} (bnum = (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e'))))\000C))))
By
Latex:
(SimpEqPairs THEN RepeatFor 2 (ParallelOp -5) THEN Auto)
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