Proof of Nuprl Lemma : pv11_p1_scout_state_subset

Step * 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

⊢ ↓l_subset(pv11_p1_Ballot_Num() × ℤ × Cmd;snd(snd(snd(msgval(e))));snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es;e)))

BY
{ ((RWO "pv11_p1_scout_state_fun_eq" 0 THENA Auto) 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

⊢ ↓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))))

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. bnum : pv11_p1_Ballot_Num()@i
16. accpts : bag(Id)@i
17. ldrs : bag(Id)@i
18. ldrs_uid : Id ⟶ ℤ@i
19. reps : bag(Id)@i
20. Inj(Id;ℤ;ldrs_uid)@i
21. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
22. header(e) = ``pv11_p1 p1b`` ∈ Name@i
23. has-es-info-type(es;e;f;Id
× pv11_p1_Ballot_Num()
× pv11_p1_Ballot_Num()
× ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))@i
24. (fst(snd(msgval(e)))) = bnum ∈ pv11_p1_Ballot_Num()@i
25. (fst(snd(snd(msgval(e))))) = bnum ∈ pv11_p1_Ballot_Num()@i
26. ↑e ∈_b pv11_p1_p1b'base(Cmd;f)
⊢ ↓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_ScoutStateFun(Cmd;accpts;f;bnum;es;pred(e)))))

3
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. ¬↑e ∈_b pv11_p1_p1b'base(Cmd;f)
15. bnum : pv11_p1_Ballot_Num()@i
16. accpts : bag(Id)@i
17. ldrs : bag(Id)@i
18. ldrs_uid : Id ⟶ ℤ@i
19. reps : bag(Id)@i
20. Inj(Id;ℤ;ldrs_uid)@i
21. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
22. header(e) = ``pv11_p1 p1b`` ∈ Name@i
23. has-es-info-type(es;e;f;Id
× pv11_p1_Ballot_Num()
× pv11_p1_Ballot_Num()
× ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))@i
24. (fst(snd(msgval(e)))) = bnum ∈ pv11_p1_Ballot_Num()@i
25. (fst(snd(snd(msgval(e))))) = bnum ∈ pv11_p1_Ballot_Num()@i
26. ↑first(e)
⊢ ↓l_subset(pv11_p1_Ballot_Num() × ℤ × Cmd;snd(snd(snd(msgval(e))));snd(pv11_p1_init_scout(Cmd;accpts)))

4
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_p1b'base(Cmd;f)
16. bnum : pv11_p1_Ballot_Num()@i
17. accpts : bag(Id)@i
18. ldrs : bag(Id)@i
19. ldrs_uid : Id ⟶ ℤ@i
20. reps : bag(Id)@i
21. Inj(Id;ℤ;ldrs_uid)@i
22. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
23. header(e) = ``pv11_p1 p1b`` ∈ Name@i
24. has-es-info-type(es;e;f;Id
× pv11_p1_Ballot_Num()
× pv11_p1_Ballot_Num()
× ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))@i
25. (fst(snd(msgval(e)))) = bnum ∈ pv11_p1_Ballot_Num()@i
26. (fst(snd(snd(msgval(e))))) = bnum ∈ pv11_p1_Ballot_Num()@i
⊢ ↓l_subset(pv11_p1_Ballot_Num()
× ℤ
× Cmd;snd(snd(snd(msgval(e))));snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;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. 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 \mvdash{} \mdownarrow{}l\_subset(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd;snd(snd(snd(msgval(e))));snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;bnum;es;e))) By Latex: ((RWO "pv11\_p1\_scout\_state\_fun\_eq" 0 THENA Auto) THEN Repeat (AutoSplit)) Home Index