Step
*
1
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. p : ℤ × Cmd@i
22. (p ∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e))))@i
⊢ pv11_p1_valid-proposal(Cmd;es;e;p;f)
BY
{ (D -2
THEN (FLemma `pv11_p1_ldr_fun_proposal3` [-1] THENA Auto)
THEN ExRepD
THEN D -1
THEN ExRepD
THEN Try (Complete (((D 0 THEN Auto) THEN With ⌈e'⌉ (D 0)⋅ THEN Auto)⋅))
THEN RepeatFor 2 (RW ClassRelStepC (-3))
THEN Auto
THEN D (-3)
THEN FHyp 19 [-5]
THEN Auto
THEN UnfoldTopAb (-1)
THEN (RevHypSubst (-4) (-1) THENA Auto)
THEN Reduce (-1)
THEN UnfoldTopAb (-1)
THEN (RWO "l_all_iff" (-1) THENA Auto)
THEN With ⌈<p1, p2>⌉ (D (-1))⋅
THEN Auto
THEN (D -1 THENA Auto)
THEN (Assert e' c≤ e BY
Auto)
THEN FLemma `pv11_p1_valid-proposal-transitivity` [-2;-1]
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. pv11\_p1\_valid-AcceptorState(Cmd;ldrs$_{uid}$;es;e;f)
21. p : \mBbbZ{} \mtimes{} Cmd@i
22. (p \mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e))))@i
\mvdash{} pv11\_p1\_valid-proposal(Cmd;es;e;p;f)
By
Latex:
(D -2
THEN (FLemma `pv11\_p1\_ldr\_fun\_proposal3` [-1] THENA Auto)
THEN ExRepD
THEN D -1
THEN ExRepD
THEN Try (Complete (((D 0 THEN Auto) THEN With \mkleeneopen{}e'\mkleeneclose{} (D 0)\mcdot{} THEN Auto)\mcdot{}))
THEN RepeatFor 2 (RW ClassRelStepC (-3))
THEN Auto
THEN D (-3)
THEN FHyp 19 [-5]
THEN Auto
THEN UnfoldTopAb (-1)
THEN (RevHypSubst (-4) (-1) THENA Auto)
THEN Reduce (-1)
THEN UnfoldTopAb (-1)
THEN (RWO "l\_all\_iff" (-1) THENA Auto)
THEN With \mkleeneopen{}<p1, p2>\mkleeneclose{} (D (-1))\mcdot{}
THEN Auto
THEN (D -1 THENA Auto)
THEN (Assert e' c\mleq{} e BY
Auto)
THEN FLemma `pv11\_p1\_valid-proposal-transitivity` [-2;-1]
THEN Auto)
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