Proof of Nuprl Lemma : pv11_p1_acc_rcv

Step * 1 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. accpts : bag(Id)@i
15. ldrs : bag(Id)@i
16. ldrs_uid : Id ─→ ℤ@i
17. reps : bag(Id)@i
18. b : pv11_p1_Ballot_Num()@i
19. s : ℤ@i
20. c : Cmd@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 p2a`` ∈ Name@i
24. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)@i

25. ∃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

26. (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e))) = (fst(snd(msgval(e)))) ∈ pv11_p1_Ballot_Num()@i
⊢ ↓(snd(msgval(e)) ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)))
BY
{ (MoveToConcl (-1)
THEN (Assert ⌈↑e ∈_b pv11_p1_p2a'base(Cmd;f)⌉⋅

         THENA (RepUR ``pv11_p1_p2a'base`` 0 THEN BLemma `member-base-class_iff` THEN Auto THEN RWO "-3" 0 THEN Auto)

         )
   THEN (RWO "pv11_p1_acc_state_fun_eq2" 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

25. ∃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

26. ↑e ∈_b pv11_p1_p2a'base(Cmd;f)
27. ↑e ∈_b pv11_p1_p1a'base(Cmd;f)
28. ↑first(e)
⊢ ((fst((pv11_p1_on_p1a(Cmd;ldrs_uid) loc(e) pv11_p1_p1a'base(Cmd;f)@e pv11_p1_init_acceptor(Cmd))))
= (fst(snd(msgval(e))))
∈ pv11_p1_Ballot_Num())
⇒

 (↓(snd(msgval(e)) ∈ snd((pv11_p1_on_p1a(Cmd;ldrs_uid) loc(e) pv11_p1_p1a'base(Cmd;f)@e pv11_p1_init_acceptor(Cmd)))))

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. accpts : bag(Id)@i
16. ldrs : bag(Id)@i
17. ldrs_uid : Id ─→ ℤ@i
18. reps : bag(Id)@i
19. b : pv11_p1_Ballot_Num()@i
20. s : ℤ@i
21. c : Cmd@i
22. Inj(Id;ℤ;ldrs_uid)@i
23. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
24. header(e) = ``pv11_p1 p2a`` ∈ Name@i
25. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)@i

26. ∃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

27. ↑e ∈_b pv11_p1_p2a'base(Cmd;f)
28. ↑e ∈_b pv11_p1_p1a'base(Cmd;f)
⊢ ((fst((pv11_p1_on_p1a(Cmd;ldrs_uid) loc(e) pv11_p1_p1a'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_p1a(Cmd;ldrs_uid) loc(e) pv11_p1_p1a'base(Cmd;f)@e
pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;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_p1a'base(Cmd;f)
15. accpts : bag(Id)@i
16. ldrs : bag(Id)@i
17. ldrs_uid : Id ─→ ℤ@i
18. reps : bag(Id)@i
19. b : pv11_p1_Ballot_Num()@i
20. s : ℤ@i
21. c : Cmd@i
22. Inj(Id;ℤ;ldrs_uid)@i
23. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
24. header(e) = ``pv11_p1 p2a`` ∈ Name@i
25. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)@i

26. ∃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

27. ↑e ∈_b pv11_p1_p2a'base(Cmd;f)
28. ↑e ∈_b pv11_p1_p2a'base(Cmd;f)
29. ↑first(e)
⊢ ((fst((pv11_p1_on_p2a(Cmd;ldrs_uid) loc(e) pv11_p1_p2a'base(Cmd;f)@e pv11_p1_init_acceptor(Cmd))))
= (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_init_acceptor(Cmd)))))

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_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))))))

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. accpts : bag(Id)@i 15. ldrs : bag(Id)@i 16. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i 17. reps : bag(Id)@i 18. b : pv11\_p1\_Ballot\_Num()@i 19. s : \mBbbZ{}@i 20. c : Cmd@i 21. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i 22. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i 23. header(e) = ``pv11\_p1 p2a``@i 24. has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd)@i 25. \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 26. (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e))) = (fst(snd(msgval(e\000C))))@i \mvdash{} \mdownarrow{}(snd(msgval(e)) \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e))) By Latex: (MoveToConcl (-1) THEN (Assert \mkleeneopen{}\muparrow{}e \mmember{}\msubb{} pv11\_p1\_p2a'base(Cmd;f)\mkleeneclose{}\mcdot{} THENA (RepUR ``pv11\_p1\_p2a'base`` 0 THEN BLemma `member-base-class\_iff` THEN Auto THEN RWO "-3" 0 THEN Auto) ) THEN (RWO "pv11\_p1\_acc\_state\_fun\_eq2" 0 THENA Auto) THEN Repeat (AutoSplit)) Home Index