Proof of Nuprl Lemma : pv11_p1_ldr_state_propose

Step * 1 1 of Lemma pv11_p1_ldr_state_propose_pred

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

⊢ <bnum, active, pv11_p1_add_if_new() pv11_p1_same_proposal(Cmd) <s, c> proposals> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)\000C(e)

BY
{ ((RWO "pv11_p1_ldr_state_eq2" 0 THENA Auto)
THEN Repeat (AutoSplit)
THEN Try (Complete ((UnfoldAtAddr [1;3] (-1) THEN FLemma `member-base-class` [-1]⋅ 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. e1 : E@i
14. e2 : E@i
15. ldrs_uid : Id ─→ ℤ@i
16. bnum : pv11_p1_Ballot_Num()@i
17. active : 𝔹@i
18. proposals : (ℤ × Cmd) List@i
19. s : ℤ@i
20. c : Cmd@i
21. (e1 <loc e2)@i
22. <bnum, active, proposals> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e1)@i
23. header(e1) = [propose] ∈ Name@i
24. has-es-info-type(es;e1;f;ℤ × Cmd)@i
25. <s, c> = msgval(e1) ∈ (ℤ × Cmd)@i
26. ¬↑(pv11_p1_in_domain(Cmd) s proposals)@i
27. e : {e:E| ¬↑first(e)}
28. ¬↑first(e)
29. e1 = pred(e) ∈ E
30. e ≤loc e2
31. e1 = pred(e) ∈ E
32. e ≤loc e2
33. ↑pred(e) ∈_b pv11_p1_propose'base(Cmd;f)
⊢ ∃x:ℤ × Cmd
   ∃s1:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)
    (x ∈ pv11_p1_propose'base(Cmd;f)(pred(e))
    ∧ s1 ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
    ∧ (<bnum, active, pv11_p1_add_if_new() pv11_p1_same_proposal(Cmd) <s, c> proposals>
      = (pv11_p1_on_propose(Cmd) loc(e) x s1)
      ∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List))))

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. e1 : E@i
14. e2 : E@i
15. ldrs_uid : Id ─→ ℤ@i
16. bnum : pv11_p1_Ballot_Num()@i
17. active : 𝔹@i
18. proposals : (ℤ × Cmd) List@i
19. s : ℤ@i
20. c : Cmd@i
21. (e1 <loc e2)@i
22. <bnum, active, proposals> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e1)@i
23. header(e1) = [propose] ∈ Name@i
24. has-es-info-type(es;e1;f;ℤ × Cmd)@i
25. <s, c> = msgval(e1) ∈ (ℤ × Cmd)@i
26. ¬↑(pv11_p1_in_domain(Cmd) s proposals)@i
27. e : {e:E| ¬↑first(e)}
28. ¬↑pred(e) ∈_b pv11_p1_preempted'base(Cmd;f)
29. ¬↑pred(e) ∈_b pv11_p1_adopted'base(Cmd;f)
30. ¬↑pred(e) ∈_b pv11_p1_propose'base(Cmd;f)
31. ¬↑first(e)
32. e1 = pred(e) ∈ E
33. e ≤loc e2
34. e1 = pred(e) ∈ E
35. e ≤loc e2

⊢ <bnum, active, pv11_p1_add_if_new() pv11_p1_same_proposal(Cmd) <s, c> proposals> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)\000C(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. e1 : E@i 14. e2 : E@i 15. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i 16. bnum : pv11\_p1\_Ballot\_Num()@i 17. active : \mBbbB{}@i 18. proposals : (\mBbbZ{} \mtimes{} Cmd) List@i 19. s : \mBbbZ{}@i 20. c : Cmd@i 21. (e1 <loc e2)@i 22. <bnum, active, proposals> \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e1)@i 23. header(e1) = [propose]@i 24. has-es-info-type(es;e1;f;\mBbbZ{} \mtimes{} Cmd)@i 25. <s, c> = msgval(e1)@i 26. \mneg{}\muparrow{}(pv11\_p1\_in\_domain(Cmd) s proposals)@i 27. e : \{e:E| \mneg{}\muparrow{}first(e)\} 28. e1 = pred(e) 29. e \mleq{}loc e2 30. e1 = pred(e) 31. e \mleq{}loc e2 \mvdash{} <bnum, active, pv11\_p1\_add\_if\_new() pv11\_p1\_same\_proposal(Cmd) <s, c> proposals> \mmember{} pv11\_p1\_LeaderS\000Ctate(Cmd;ldrs$_{uid}$;f)(e) By Latex: ((RWO "pv11\_p1\_ldr\_state\_eq2" 0 THENA Auto) THEN Repeat (AutoSplit) THEN Try (Complete ((UnfoldAtAddr [1;3] (-1) THEN FLemma `member-base-class` [-1]\mcdot{} THEN Auto)))) Home Index