Proof of Nuprl Lemma : pv11

Step * 1 1 of Lemma pv11_p1-validity2

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. p1 : ℤ × Cmd@i
20. p1 ∈ pv11_p1_decision'base(Cmd;f)(e)@i
21. ∀e:E. ∀i:Id. ∀p:ℤ × Cmd.
(pv11_p1_decision'send(Cmd;f) i p ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e)
⇒ pv11_p1_valid-proposal(Cmd;es;e;p;f))

⊢ ↓∃e':E. ((e' < e) ∧ (msg-header(info(e')) = [propose] ∈ Name) ∧ (p1 = msg-body(info(e')) ∈ (ℤ × Cmd)))

BY
{ (DProdsAndUnions
THEN (UseMessageConstraint' (-2)⋅ THENA Auto)
THEN (Subst ⌈delay = 0 ∈ ℤ⌉ (-1)⋅

         THENA (Auto THEN Try (ProveEncodesMsgType) THEN RewriteWithILF (-1) THEN Auto THEN ExRepD 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. 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. p2 : ℤ@i
20. p3 : Cmd@i
21. <p2, p3> ∈ pv11_p1_decision'base(Cmd;f)(e)@i
22. ∀e:E. ∀i:Id. ∀p:ℤ × Cmd.
(pv11_p1_decision'send(Cmd;f) i p ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e)
⇒ pv11_p1_valid-proposal(Cmd;es;e;p;f))
23. e' : E
24. delay : ℤ
25. (e' < e)

26. make-msg-interface(0;loc(e);mk-msg(msg-authentic(info(e));[decision];<p2, p3>)) ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_\000Cuid;reps;f)(e')

⊢ ↓∃e':E. ((e' < e) ∧ (msg-header(info(e')) = [propose] ∈ Name) ∧ (<p2, p3> = msg-body(info(e')) ∈ (ℤ × Cmd)))

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. p1 : \mBbbZ{} \mtimes{} Cmd@i 20. p1 \mmember{} pv11\_p1\_decision'base(Cmd;f)(e)@i 21. \mforall{}e:E. \mforall{}i:Id. \mforall{}p:\mBbbZ{} \mtimes{} Cmd. (pv11\_p1\_decision'send(Cmd;f) i p \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid}\mbackslash{}f\000Cf24;reps;f)(e) {}\mRightarrow{} pv11\_p1\_valid-proposal(Cmd;es;e;p;f)) \mvdash{} \mdownarrow{}\mexists{}e':E. ((e' < e) \mwedge{} (msg-header(info(e')) = [propose]) \mwedge{} (p1 = msg-body(info(e')))) By Latex: (DProdsAndUnions THEN (UseMessageConstraint' (-2)\mcdot{} THENA Auto) THEN (Subst \mkleeneopen{}delay = 0\mkleeneclose{} (-1)\mcdot{} THENA (Auto THEN Try (ProveEncodesMsgType) THEN RewriteWithILF (-1) THEN Auto THEN ExRepD THEN Auto) )) Home Index