Step
*
1
1
1
1
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. header(e) = ``pv11_p1 p2a`` ∈ Name
21. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
22. ∀e:E
((header(e) ∈ pv11_p1_headers_no_inputs())
⇒ (↓∃e':E
∃delay:ℤ
((e' < e) ∧ make-msg-interface(delay;loc(e);info(e)) ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e'))))
23. e' : E
24. delay : ℤ
25. (e' < e)
26. loc(e') ↓∈ ldrs
27. loc(e) ↓∈ accpts
28. delay = 0 ∈ ℤ
29. es-info-auth(es;e) = pv11_p1_init_active()
30. (fst(msgval(e))) = loc(e') ∈ Id
31. header(e') = [propose] ∈ Name
32. has-es-info-type(es;e';f;ℤ × Cmd)
33. ↑(fst(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'))))
34. ¬↑(pv11_p1_in_domain(Cmd) (fst(msgval(e'))) (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e')))))
35. (fst(snd(msgval(e)))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num()
36. <fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> = msgval(e') ∈ (ℤ × Cmd)
⊢ pv11_p1_valid-proposal(Cmd;es;e;snd(snd(<fst(msgval(e))
, fst(snd(msgval(e)))
, fst(snd(snd(msgval(e))))
, snd(snd(snd(msgval(e))))>));f)
BY
{ (D 0
THEN Reduce 0
THEN Auto
THEN With ⌈e'⌉ (D 0)⋅
THEN Auto
THEN RepeatFor 2 (RW ClassRelStepC 0)
THEN Auto
THEN D 0
THEN Auto
THEN SimpleSubstVarAll 0
THEN (At ⌈Type⌉ (SimpleSubstTermAtAddr [2;1] 0)⋅ THENA Auto)
THEN (At ⌈Type⌉ (SimpleSubstTermAtAddr [2;2] 0)⋅ THENA Auto)
THEN RWO "pair-eta<" 0
THEN Auto) }
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. header(e) = ``pv11\_p1 p2a``
21. has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd)
22. \mforall{}e:E
((header(e) \mmember{} pv11\_p1\_headers\_no\_inputs())
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E
\mexists{}delay:\mBbbZ{}
((e' < e)
\mwedge{} make-msg-interface(delay;loc(e);info(e)) \mmember{}
pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid}$;reps;f)(e'))))
23. e' : E
24. delay : \mBbbZ{}
25. (e' < e)
26. loc(e') \mdownarrow{}\mmember{} ldrs
27. loc(e) \mdownarrow{}\mmember{} accpts
28. delay = 0
29. es-info-auth(es;e) = pv11\_p1\_init\_active()
30. (fst(msgval(e))) = loc(e')
31. header(e') = [propose]
32. has-es-info-type(es;e';f;\mBbbZ{} \mtimes{} Cmd)
33. \muparrow{}(fst(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e'))))
34. \mneg{}\muparrow{}(pv11\_p1\_in\_domain(Cmd) (fst(msgval(e')))
(snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e')))))
35. (fst(snd(msgval(e)))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e'\000C)))
36. <fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> = msgval(e')
\mvdash{} pv11\_p1\_valid-proposal(Cmd;es;e;snd(snd(<fst(msgval(e))
, fst(snd(msgval(e)))
, fst(snd(snd(msgval(e))))
, snd(snd(snd(msgval(e))))>));f)
By
Latex:
(D 0
THEN Reduce 0
THEN Auto
THEN With \mkleeneopen{}e'\mkleeneclose{} (D 0)\mcdot{}
THEN Auto
THEN RepeatFor 2 (RW ClassRelStepC 0)
THEN Auto
THEN D 0
THEN Auto
THEN SimpleSubstVarAll 0
THEN (At \mkleeneopen{}Type\mkleeneclose{} (SimpleSubstTermAtAddr [2;1] 0)\mcdot{} THENA Auto)
THEN (At \mkleeneopen{}Type\mkleeneclose{} (SimpleSubstTermAtAddr [2;2] 0)\mcdot{} THENA Auto)
THEN RWO "pair-eta<" 0
THEN Auto)
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