Proof of Nuprl Lemma : pv11

Step * 1 of Lemma pv11_p1-validity

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

{ (InstLemma `pv11_p1_validity-lemma` [⌈Cmd⌉;⌈f⌉;⌈es⌉;⌈accpts⌉;⌈ldrs⌉;⌈ldrs_uid⌉;⌈reps⌉]⋅

   THEN Auto
   THEN RepUR ``pv11_p1_decision'send`` (-2)
   THEN DProdsAndUnions
   THEN (RWO "make-Msg-as-mk-msg" (-2) THENA Auto)
   THEN (RewriteWithILF (-2) THENA Auto)
   THEN SqExRepD
   THEN DProdsAndUnions) }

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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs_uid : Id ─→ ℤ@i
16. reps : bag(Id)@i
17. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
18. e : E@i
19. i : Id@i
20. p1 : ℤ@i
21. p2 : Cmd@i
22. loc(e) ↓∈ ldrs
23. e' : {e':E| e' ≤loc e }
24. header(e) = ``pv11_p1 p2b`` ∈ Name
25. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
26. 0 = 0 ∈ ℤ
27. (fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e))))) ∈ pv11_p1_Ballot_Num()
28. #(pv11_p1_CommanderStateFun(Cmd;accpts;f;fst(snd(msgval(e)));fst(snd(snd(msgval(e))));es.e';e)) < ...
29. i ↓∈ reps
30. ff = pv11_p1_init_active()
31. p1 = (fst(snd(snd(msgval(e))))) ∈ ℤ
32. header(e') = [propose] ∈ Name
33. has-es-info-type(es;e';f;ℤ × Cmd)
34. ↑(fst(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'))))
35. ¬↑(pv11_p1_in_domain(Cmd) (fst(msgval(e'))) (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e')))))
36. (fst(snd(msgval(e)))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num()
37. <fst(snd(snd(msgval(e)))), p2> = msgval(e') ∈ (ℤ × Cmd)

38. (no ((pv11_p1_commander_output(Cmd;accpts;reps;f) <fst(snd(msgval(e))), fst(snd(snd(msgval(e)))), p2> o pv11_p1_p2b'\000Cbase(Cmd;f)) o

        pv11_p1_CommanderState(Cmd;accpts;f) (fst(snd(msgval(e)))) (fst(snd(snd(msgval(e)))))) between e' and e)

39. ∀e:E
      (pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e;f)
      ∧ pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e;f)
      ∧ (∀b:pv11_p1_Ballot_Num(). ∀start:E.  (start ≤loc e  ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e;f)))
      ∧ ((header(e) = ``pv11_p1 adopted`` ∈ Name)
        ⇒ has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
        ⇒ pv11_p1_valid-adopted-message(Cmd;es;e;f))
      ∧ ((header(e) = ``pv11_p1 p1b`` ∈ Name)
        ⇒ has-es-info-type(es;e;f;Id
           × pv11_p1_Ballot_Num()
           × pv11_p1_Ballot_Num()
           × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
        ⇒ pv11_p1_valid-p1b-message(Cmd;es;e;f))
      ∧ ((header(e) = ``pv11_p1 p2a`` ∈ Name)
        ⇒ has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
        ⇒ pv11_p1_valid-p2a-message(Cmd;es;e;f)))
⊢ pv11_p1_valid-proposal(Cmd;es;e;<p1, p2>;f)

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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs_uid : Id ─→ ℤ@i
16. reps : bag(Id)@i
17. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
18. e : E@i
19. i : Id@i
20. p1 : ℤ@i
21. p2 : Cmd@i
22. loc(e) ↓∈ ldrs
23. e' : {e':E| e' ≤loc e }
24. header(e) = ``pv11_p1 p2b`` ∈ Name
25. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
26. 0 = 0 ∈ ℤ
27. (fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e))))) ∈ pv11_p1_Ballot_Num()
28. #(pv11_p1_CommanderStateFun(Cmd;accpts;f;fst(snd(msgval(e)));fst(snd(snd(msgval(e))));es.e';e)) < ...
29. i ↓∈ reps
30. ff = pv11_p1_init_active()
31. p1 = (fst(snd(snd(msgval(e))))) ∈ ℤ
32. header(e') = ``pv11_p1 adopted`` ∈ Name
33. has-es-info-type(es;e';f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
34. (fst(msgval(e'))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num()
35. <fst(snd(snd(msgval(e)))), p2> ↓∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e')))
36. ¬(∃p2:Cmd. (<fst(snd(snd(msgval(e)))), p2> ∈ pv11_p1_pmax(Cmd;ldrs_uid) (snd(msgval(e')))))
37. (fst(snd(msgval(e)))) = (fst(msgval(e'))) ∈ pv11_p1_Ballot_Num()

38. (no ((pv11_p1_commander_output(Cmd;accpts;reps;f) <fst(snd(msgval(e))), fst(snd(snd(msgval(e)))), p2> o pv11_p1_p2b'\000Cbase(Cmd;f)) o

        pv11_p1_CommanderState(Cmd;accpts;f) (fst(snd(msgval(e)))) (fst(snd(snd(msgval(e)))))) between e' and e)

39. ∀e:E
      (pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e;f)
      ∧ pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e;f)
      ∧ (∀b:pv11_p1_Ballot_Num(). ∀start:E.  (start ≤loc e  ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e;f)))
      ∧ ((header(e) = ``pv11_p1 adopted`` ∈ Name)
        ⇒ has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
        ⇒ pv11_p1_valid-adopted-message(Cmd;es;e;f))
      ∧ ((header(e) = ``pv11_p1 p1b`` ∈ Name)
        ⇒ has-es-info-type(es;e;f;Id
           × pv11_p1_Ballot_Num()
           × pv11_p1_Ballot_Num()
           × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
        ⇒ pv11_p1_valid-p1b-message(Cmd;es;e;f))
      ∧ ((header(e) = ``pv11_p1 p2a`` ∈ Name)
        ⇒ has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
        ⇒ pv11_p1_valid-p2a-message(Cmd;es;e;f)))
⊢ pv11_p1_valid-proposal(Cmd;es;e;<p1, p2>;f)

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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs_uid : Id ─→ ℤ@i
16. reps : bag(Id)@i
17. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
18. e : E@i
19. i : Id@i
20. p1 : ℤ@i
21. p2 : Cmd@i
22. loc(e) ↓∈ ldrs
23. e' : {e':E| e' ≤loc e }
24. header(e) = ``pv11_p1 p2b`` ∈ Name
25. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
26. 0 = 0 ∈ ℤ
27. (fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e))))) ∈ pv11_p1_Ballot_Num()
28. #(pv11_p1_CommanderStateFun(Cmd;accpts;f;fst(snd(msgval(e)));fst(snd(snd(msgval(e))));es.e';e)) < ...
29. i ↓∈ reps
30. ff = pv11_p1_init_active()
31. p1 = (fst(snd(snd(msgval(e))))) ∈ ℤ
32. header(e') = ``pv11_p1 adopted`` ∈ Name
33. has-es-info-type(es;e';f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
34. (fst(msgval(e'))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num()
35. v2 : pv11_p1_Ballot_Num()
36. <v2, fst(snd(snd(msgval(e)))), p2> ↓∈ snd(msgval(e'))

37. ¬(∃z5:pv11_p1_Ballot_Num(). ∃z8:Cmd. ((↑(v2  < z5)) ∧ (<z5, fst(snd(snd(msgval(e)))), z8> ∈ snd(msgval(e')))))

38. (fst(snd(msgval(e)))) = (fst(msgval(e'))) ∈ pv11_p1_Ballot_Num()

39. (no ((pv11_p1_commander_output(Cmd;accpts;reps;f) <fst(snd(msgval(e))), fst(snd(snd(msgval(e)))), p2> o pv11_p1_p2b'\000Cbase(Cmd;f)) o

        pv11_p1_CommanderState(Cmd;accpts;f) (fst(snd(msgval(e)))) (fst(snd(snd(msgval(e)))))) between e' and e)

40. ∀e:E
      (pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e;f)
      ∧ pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e;f)
      ∧ (∀b:pv11_p1_Ballot_Num(). ∀start:E.  (start ≤loc e  ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e;f)))
      ∧ ((header(e) = ``pv11_p1 adopted`` ∈ Name)
        ⇒ has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
        ⇒ pv11_p1_valid-adopted-message(Cmd;es;e;f))
      ∧ ((header(e) = ``pv11_p1 p1b`` ∈ Name)
        ⇒ has-es-info-type(es;e;f;Id
           × pv11_p1_Ballot_Num()
           × pv11_p1_Ballot_Num()
           × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
        ⇒ pv11_p1_valid-p1b-message(Cmd;es;e;f))
      ∧ ((header(e) = ``pv11_p1 p2a`` ∈ Name)
        ⇒ has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
        ⇒ pv11_p1_valid-p2a-message(Cmd;es;e;f)))
⊢ pv11_p1_valid-proposal(Cmd;es;e;<p1, p2>;f)

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. accpts : bag(Id)@i 14. ldrs : bag(Id)@i 15. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i 16. reps : bag(Id)@i 17. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i 18. e : E@i 19. i : Id@i 20. p : \mBbbZ{} \mtimes{} Cmd@i 21. pv11\_p1\_decision'send(Cmd;f) i p \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid}$\000C;reps;f)(e)@i \mvdash{} pv11\_p1\_valid-proposal(Cmd;es;e;p;f) By Latex: (InstLemma `pv11\_p1\_validity-lemma` [\mkleeneopen{}Cmd\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}accpts\mkleeneclose{};\mkleeneopen{}ldrs\mkleeneclose{};\mkleeneopen{}ldrs$_{uid}\mbackslash{}ff2\000C4\mkleeneclose{};\mkleeneopen{}reps\mkleeneclose{}]\mcdot{} THEN Auto THEN RepUR ``pv11\_p1\_decision'send`` (-2) THEN DProdsAndUnions THEN (RWO "make-Msg-as-mk-msg" (-2) THENA Auto) THEN (RewriteWithILF (-2) THENA Auto) THEN SqExRepD THEN DProdsAndUnions) Home Index