Proof of Nuprl Lemma : pv11_p1

Step * 1 3 2 of Lemma pv11_p1_validity-lemma

1. Cmd : {T:Type| valueall-type(T)}
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. ∀e1:E
      ((e1 < e)
      ⇒ (pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e1;f)
         ∧ pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e1;f)
         ∧ (∀b:pv11_p1_Ballot_Num(). ∀start:E.
              (start ≤loc e1  ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e1;f)))
         ∧ ((header(e1) = ``pv11_p1 adopted`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
           ⇒ pv11_p1_valid-adopted-message(Cmd;es;e1;f))
         ∧ ((header(e1) = ``pv11_p1 p1b`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;Id
              × pv11_p1_Ballot_Num()
              × pv11_p1_Ballot_Num()
              × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
           ⇒ pv11_p1_valid-p1b-message(Cmd;es;e1;f))
         ∧ ((header(e1) = ``pv11_p1 p2a`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
           ⇒ pv11_p1_valid-p2a-message(Cmd;es;e1;f))))
20. pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e;f)
21. pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e;f)
22. b : pv11_p1_Ballot_Num()@i
23. start : E@i
24. start ≤loc e @i
25. p : ℤ × Cmd@i
26. y : pv11_p1_Ballot_Num() × ℤ × Cmd
27. (y ∈ snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;b;es.start;e)))
28. p = (snd(y)) ∈ (ℤ × Cmd)
29. e' : E
30. l : Id
31. r : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
32. e' ≤loc e
33. (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))

∧ (<l, b, b, r>
  = msgval(e')
  ∈ (Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
34. (y ∈ r)
35. pv11_p1_valid-p1b-message(Cmd;es;e';f)
⊢ pv11_p1_valid-proposal(Cmd;es;e;p;f)
BY
{ (RepD
   THEN RepUR ``pv11_p1_valid-p1b-message pv11_p1_valid-proposals`` (-1)
   THEN (InstLemma `l_all_iff` [⌈ℤ × Cmd⌉;⌈map(λp.(snd(p));snd(snd(snd(msgval(e')))))⌉;
         ⌈λ₂p.pv11_p1_valid-proposal(Cmd;es;e';p;f)⌉]⋅
         THENA Auto
         )
   THEN RWO "-1" (-2)
   THEN Thin (-1)
   THEN (With ⌈p⌉ (D (-1))⋅ THENA Auto)
   THEN (Assert e' c≤ e BY
               OnMaybeHyp 32 (\h. (RWO  "eo-forward-le" h THEN CompleteAuto)))
   THEN (Using [`e',⌈e'⌉] (BLemma `pv11_p1_valid-proposal-transitivity`)⋅ THENA Auto)
   THEN Try (Trivial)) }

1
1. Cmd : {T:Type| valueall-type(T)}
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

34. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))

35. <l, b, b, r> = msgval(e') ∈ (Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) Li\000Cst))

36. (y ∈ r)
37. (p ∈ map(λp.(snd(p));snd(snd(snd(msgval(e')))))) ⇒ pv11_p1_valid-proposal(Cmd;es;e';p;f)
38. e' c≤ e
⊢ pv11_p1_valid-proposal(Cmd;es;e';p;f)

Latex:

Latex:
1. Cmd : \{T:Type| valueall-type(T)\} 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. \mforall{}e1:E ((e1 < e) {}\mRightarrow{} (pv11\_p1\_valid-AcceptorState(Cmd;ldrs$_{uid}$;es;e1;f) \mwedge{} pv11\_p1\_valid-LeaderState(Cmd;ldrs$_{uid}$;es;e1;f) \mwedge{} (\mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}start:E. (start \mleq{}loc e1 {}\mRightarrow{} pv11\_p1\_valid-ScoutState(Cmd;accpts;b;start;es;e1;f))) \mwedge{} ((header(e1) = ``pv11\_p1 adopted``) {}\mRightarrow{} has-es-info-type(es;e1;f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)) {}\mRightarrow{} pv11\_p1\_valid-adopted-message(Cmd;es;e1;f)) \mwedge{} ((header(e1) = ``pv11\_p1 p1b``) {}\mRightarrow{} has-es-info-type(es;e1;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)) {}\mRightarrow{} pv11\_p1\_valid-p1b-message(Cmd;es;e1;f)) \mwedge{} ((header(e1) = ``pv11\_p1 p2a``) {}\mRightarrow{} has-es-info-type(es;e1;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) {}\mRightarrow{} pv11\_p1\_valid-p2a-message(Cmd;es;e1;f)))) 20. pv11\_p1\_valid-AcceptorState(Cmd;ldrs$_{uid}$;es;e;f) 21. pv11\_p1\_valid-LeaderState(Cmd;ldrs$_{uid}$;es;e;f) 22. b : pv11\_p1\_Ballot\_Num()@i 23. start : E@i 24. start \mleq{}loc e @i 25. p : \mBbbZ{} \mtimes{} Cmd@i 26. y : pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd 27. (y \mmember{} snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;b;es.start;e))) 28. p = (snd(y)) 29. e' : E 30. l : Id 31. r : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List 32. e' \mleq{}loc e 33. (header(e') = ``pv11\_p1 p1b``) \mwedge{} 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)) \mwedge{} (<l, b, b, r> = msgval(e')) 34. (y \mmember{} r) 35. pv11\_p1\_valid-p1b-message(Cmd;es;e';f) \mvdash{} pv11\_p1\_valid-proposal(Cmd;es;e;p;f) By Latex: (RepD THEN RepUR ``pv11\_p1\_valid-p1b-message pv11\_p1\_valid-proposals`` (-1) THEN (InstLemma `l\_all\_iff` [\mkleeneopen{}\mBbbZ{} \mtimes{} Cmd\mkleeneclose{};\mkleeneopen{}map(\mlambda{}p.(snd(p));snd(snd(snd(msgval(e')))))\mkleeneclose{}; \mkleeneopen{}\mlambda{}\msubtwo{}p.pv11\_p1\_valid-proposal(Cmd;es;e';p;f)\mkleeneclose{}]\mcdot{} THENA Auto ) THEN RWO "-1" (-2) THEN Thin (-1) THEN (With \mkleeneopen{}p\mkleeneclose{} (D (-1))\mcdot{} THENA Auto) THEN (Assert e' c\mleq{} e BY OnMaybeHyp 32 (\mbackslash{}h. (RWO "eo-forward-le" h THEN CompleteAuto))) THEN (Using [`e',\mkleeneopen{}e'\mkleeneclose{}] (BLemma `pv11\_p1\_valid-proposal-transitivity`)\mcdot{} THENA Auto) THEN Try (Trivial)) Home Index