Step
*
1
1
of Lemma
pv11_p1_scout_state_subset_pvals
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. bnum : pv11_p1_Ballot_Num()@i
15. accpts : bag(Id)@i
16. ldrs : bag(Id)@i
17. ldrs_uid : Id ─→ ℤ@i
18. reps : bag(Id)@i
19. Inj(Id;ℤ;ldrs_uid)@i
20. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
21. header(e) = ``pv11_p1 p1b`` ∈ Name@i
22. has-es-info-type(es;e;f;Id
× pv11_p1_Ballot_Num()
× pv11_p1_Ballot_Num()
× ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))@i
23. (fst(snd(msgval(e)))) = bnum ∈ pv11_p1_Ballot_Num()@i
24. (fst(snd(snd(msgval(e))))) = bnum ∈ pv11_p1_Ballot_Num()@i
25. ↑e ∈b pv11_p1_p1b'base(Cmd;f)
26. ↑first(e)
⊢ ↓l_subset(pv11_p1_Ballot_Num() × ℤ × Cmd;snd(snd(snd(msgval(e))));snd((pv11_p1_on_p1b(Cmd) bnum loc(e)
pv11_p1_p1b'base(Cmd;f)@e
pv11_p1_init_scout(Cmd;accpts))))
BY
{ (RepUR ``pv11_p1_p1b'base`` 0⋅
THEN RWO "classfun-res-base" 0
THEN Auto
THEN GenConclAtAddr [1;3;1;1;2]
THEN DProdsAndUnions
THEN AllReduce
THEN RepUR ``pv11_p1_on_p1b pv11_p1_init_scout let`` 0
THEN Repeat (AutoSplit)) }
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. bnum : pv11_p1_Ballot_Num()@i
15. accpts : bag(Id)@i
16. ldrs : bag(Id)@i
17. ldrs_uid : Id ─→ ℤ@i
18. reps : bag(Id)@i
19. Inj(Id;ℤ;ldrs_uid)@i
20. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
21. header(e) = ``pv11_p1 p1b`` ∈ Name@i
22. has-es-info-type(es;e;f;Id
× pv11_p1_Ballot_Num()
× pv11_p1_Ballot_Num()
× ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))@i
23. (fst(snd(msgval(e)))) = bnum ∈ pv11_p1_Ballot_Num()@i
24. (fst(snd(snd(msgval(e))))) = bnum ∈ pv11_p1_Ballot_Num()@i
25. ↑e ∈b pv11_p1_p1b'base(Cmd;f)
26. ↑first(e)
27. msgval(e) ∈ Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
28. v1 : Id@i
29. v3 : pv11_p1_Ballot_Num()@i
30. v5 : pv11_p1_Ballot_Num()@i
31. v6 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
32. msgval(e)
= <v1, v3, v5, v6>
∈ (Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))@i
33. bnum = v3 ∈ pv11_p1_Ballot_Num()
34. bnum = v5 ∈ pv11_p1_Ballot_Num()
⊢ ↓l_subset(pv11_p1_Ballot_Num() × ℤ × Cmd;v6;pv11_p1_append_news(Cmd) pv11_p1_same_pvalue(Cmd)
pv11_p1_init_pvalues(Cmd)
v6)
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. bnum : pv11\_p1\_Ballot\_Num()@i
15. accpts : bag(Id)@i
16. ldrs : bag(Id)@i
17. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i
18. reps : bag(Id)@i
19. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i
20. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i
21. header(e) = ``pv11\_p1 p1b``@i
22. has-es-info-type(es;e;f;Id
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))@i
23. (fst(snd(msgval(e)))) = bnum@i
24. (fst(snd(snd(msgval(e))))) = bnum@i
25. \muparrow{}e \mmember{}\msubb{} pv11\_p1\_p1b'base(Cmd;f)
26. \muparrow{}first(e)
\mvdash{} \mdownarrow{}l\_subset(pv11\_p1\_Ballot\_Num()
\mtimes{} \mBbbZ{}
\mtimes{} Cmd;snd(snd(snd(msgval(e))));snd((pv11\_p1\_on\_p1b(Cmd) bnum loc(e) pv11\_p1\_p1b'base(Cmd;f)@e
pv11\_p1\_init\_scout(Cmd;accpts))))
By
Latex:
(RepUR ``pv11\_p1\_p1b'base`` 0\mcdot{}
THEN RWO "classfun-res-base" 0
THEN Auto
THEN GenConclAtAddr [1;3;1;1;2]
THEN DProdsAndUnions
THEN AllReduce
THEN RepUR ``pv11\_p1\_on\_p1b pv11\_p1\_init\_scout let`` 0
THEN Repeat (AutoSplit))
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