Proof of Nuprl Lemma : pv11_p1_acc

Step * of Lemma pv11_p1_acc_p2a

∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀e:E. ∀ldrs_uid:Id ─→ ℤ.

∀v:pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List). ∀b:pv11_p1_Ballot_Num(). ∀s:ℤ. ∀c:Cmd.

  (v ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e)
  ⇒ let bnum,accepted = v
     in (<b, s, c> ∈ accepted) ⇒ (∃e':E. ∃l:Id. (e' ≤loc e  ∧ <l, b, s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e'))))
BY
{ MemoryInvariant }

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. ldrs_uid : Id ─→ ℤ@i
15. v1 : pv11_p1_Ballot_Num()@i
16. v2 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
17. b : pv11_p1_Ballot_Num()@i
18. s : ℤ@i
19. c : Cmd@i
20. <v1, v2> ∈

     State-loc-comb(λloc.{pv11_p1_init_acceptor(Cmd)};pv11_p1_on_p1a(Cmd;ldrs_uid) + pv11_p1_on_p2a(Cmd;ldrs_uid);...)(

e)
21. y1 : Id@i
22. x3 : ℤ@i
23. x4 : Id@i
24. y5 : ℤ@i
25. y6 : Cmd@i
26. e' : E@i
27. x1 : ℤ@i
28. x2 : Id@i
29. s2 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
30. e' ≤loc e @i
31. inr <y1, inl <x3, x4>, y5, y6> ∈ pv11_p1_p1a'base(Cmd;f) (+) pv11_p1_p2a'base(Cmd;f)(e')@i
32. <inl <x1, x2>, s2> ∈

     State-loc-comb(λloc.{pv11_p1_init_acceptor(Cmd)};pv11_p1_on_p1a(Cmd;ldrs_uid) + pv11_p1_on_p2a(Cmd;ldrs_uid);...)(

pred(e'))@i
33. (<b, s, c> ∈ s2) ⇒ (∃e':E. ∃l:Id. (e' ≤loc e ∧ <l, b, s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e')))@i
34. x1 < x3

35. ¬(∃zzc:pv11_p1_Ballot_Num() × ℤ × Cmd. ((zzc ∈ s2) ∧ (↑(pv11_p1_same_pvalue(Cmd) <inl <x3, x4>, y5, y6> zzc))))

36. b = (inl <x3, x4>) ∈ pv11_p1_Ballot_Num()
37. s = y5 ∈ ℤ
38. c = y6 ∈ Cmd
39. ¬↑first(e')
⊢ ∃e':E. ∃l:Id. (e' ≤loc e ∧ <l, b, s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e'))

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

     State-loc-comb(λloc.{pv11_p1_init_acceptor(Cmd)};pv11_p1_on_p1a(Cmd;ldrs_uid) + pv11_p1_on_p2a(Cmd;ldrs_uid);...)(

     State-loc-comb(λloc.{pv11_p1_init_acceptor(Cmd)};pv11_p1_on_p1a(Cmd;ldrs_uid) + pv11_p1_on_p2a(Cmd;ldrs_uid);...)(

pred(e'))@i
33. (<b, s, c> ∈ s2) ⇒ (∃e':E. ∃l:Id. (e' ≤loc e ∧ <l, b, s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e')))@i
34. x1 = x3 ∈ ℤ
35. (ldrs_uid x2) ≤ (ldrs_uid x4)

36. ¬(∃zzc:pv11_p1_Ballot_Num() × ℤ × Cmd. ((zzc ∈ s2) ∧ (↑(pv11_p1_same_pvalue(Cmd) <inl <x3, x4>, y5, y6> zzc))))

37. b = (inl <x3, x4>) ∈ pv11_p1_Ballot_Num()
38. s = y5 ∈ ℤ
39. c = y6 ∈ Cmd
40. ¬↑first(e')
⊢ ∃e':E. ∃l:Id. (e' ≤loc e ∧ <l, b, s, c> ∈ pv11_p1_p2a'base(Cmd;f)(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

     State-loc-comb(λloc.{pv11_p1_init_acceptor(Cmd)};pv11_p1_on_p1a(Cmd;ldrs_uid) + pv11_p1_on_p2a(Cmd;ldrs_uid);...)(

     e)
21. y1 : Id@i
22. y3 : pv11_p1_Ballot_Num()@i
23. y5 : ℤ@i
24. y6 : Cmd@i
25. e' : E@i
26. y : 0 = 0 ∈ ℤ
27. s2 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
28. e' ≤loc e @i
29. inr <y1, y3, y5, y6>  ∈ pv11_p1_p1a'base(Cmd;f) (+) pv11_p1_p2a'base(Cmd;f)(e')@i
30. (inr Ax ) = pv11_p1_dummy_ballot() ∈ pv11_p1_Ballot_Num()
31. s2 = pv11_p1_init_accepted(Cmd) ∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
32. (<b, s, c> ∈ s2) ⇒ (∃e':E. ∃l:Id. (e' ≤loc e  ∧ <l, b, s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e')))@i

33. ¬(∃zzc:pv11_p1_Ballot_Num() × ℤ × Cmd. ((zzc ∈ s2) ∧ (↑(pv11_p1_same_pvalue(Cmd) <y3, y5, y6> zzc))))

34. b = y3 ∈ pv11_p1_Ballot_Num()
35. s = y5 ∈ ℤ
36. c = y6 ∈ Cmd
37. ↑first(e')
⊢ ∃e':E. ∃l:Id. (e' ≤loc e ∧ <l, b, s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e'))

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

     State-loc-comb(λloc.{pv11_p1_init_acceptor(Cmd)};pv11_p1_on_p1a(Cmd;ldrs_uid) + pv11_p1_on_p2a(Cmd;ldrs_uid);...)(

     State-loc-comb(λloc.{pv11_p1_init_acceptor(Cmd)};pv11_p1_on_p1a(Cmd;ldrs_uid) + pv11_p1_on_p2a(Cmd;ldrs_uid);...)(

pred(e'))@i
31. (<b, s, c> ∈ s2) ⇒ (∃e':E. ∃l:Id. (e' ≤loc e ∧ <l, b, s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e')))@i

32. ¬(∃zzc:pv11_p1_Ballot_Num() × ℤ × Cmd. ((zzc ∈ s2) ∧ (↑(pv11_p1_same_pvalue(Cmd) <y3, y5, y6> zzc))))

33. b = y3 ∈ pv11_p1_Ballot_Num()
34. s = y5 ∈ ℤ
35. c = y6 ∈ Cmd
36. ¬↑first(e')
⊢ ∃e':E. ∃l:Id. (e' ≤loc e ∧ <l, b, s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e'))

Latex:

Latex:
\mforall{}Cmd:ValueAllType. \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)). \mforall{}e:E. \mforall{}ldrs$_\mbackslash{}ff7\000Cbuid}$:Id {}\mrightarrow{} \mBbbZ{}. \mforall{}v:pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List). \mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}s:\mBbbZ{}. \mforall{}c:Cmd. (v \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e) {}\mRightarrow{} let bnum,accepted = v in (<b, s, c> \mmember{} accepted) {}\mRightarrow{} (\mexists{}e':E. \mexists{}l:Id. (e' \mleq{}loc e \mwedge{} <l, b, s, c> \mmember{} pv11\_p1\_p2a'base(Cmd;f)(e')))) By Latex: MemoryInvariant Home Index