Step
*
1
2
1
of Lemma
pv11_p1_consistency_lemma
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. Inj(Id;ℤ;ldrs_uid)@i
19. e : E@i
20. ∀e1:E
((e1 < e)
⇒ (∀b:pv11_p1_Ballot_Num(). ∀k:ℤ. ∀v:Cmd.
(pv11_p1_pvalue{i:l}(Cmd;ldrs_uid;accpts;es;e1;<b, k, v>;f)
⇒ (∀e':E. ∀w:Cmd.
(pv11_p1_decision{i:l}(Cmd;f;accpts;ldrs;ldrs_uid;reps;es;e';k;w)
⇒ has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) (snd(snd(snd(msgval(e'))))) b))
⇒ (w = v ∈ Cmd))))))
21. b : pv11_p1_Ballot_Num()@i
22. k : ℤ@i
23. v : Cmd@i
24. pv11_p1_pvalue{i:l}(Cmd;ldrs_uid;accpts;es;e;<b, k, v>;f)@i'
25. header(e) = ``pv11_p1 p2a`` ∈ Name
26. <b, k, v> = (snd(msgval(e))) ∈ (pv11_p1_Ballot_Num() × ℤ × Cmd)
27. e' : E@i
28. w : Cmd@i
29. pv11_p1_decision{i:l}(Cmd;f;accpts;ldrs;ldrs_uid;reps;es;e';k;w)@i'
30. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())@i
31. ↑(pv11_p1_leq_bnum(ldrs_uid) (snd(snd(snd(msgval(e'))))) b)@i
32. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
33. ∀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'))))
34. e1 : E
35. delay : ℤ
36. (e1 < e)
37. make-msg-interface(delay;loc(e);mk-msg(es-info-auth(es;e);``pv11_p1 p2a``;<fst(msgval(e))
, fst(snd(msgval(e)))
, fst(snd(snd(msgval(e))))
, snd(snd(snd(msgval(e))))>)) ∈ pv11_p1_ma\000Cin(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e1)@i
⊢ w = v ∈ Cmd
BY
{ ((Subst ⌈delay ~ 0⌉ (-1)⋅ THENA (RewriteWithILF (-1) THEN Auto))
THEN (Subst ⌈es-info-auth(es;e) ~ ff⌉ (-1)⋅ THENA (RewriteWithILF (-1) THEN Auto))
THEN (Subst ⌈make-msg-interface(0;loc(e);mk-msg(ff;``pv11_p1 p2a``;<fst(msgval(e))
, fst(snd(msgval(e)))
, fst(snd(snd(msgval(e))))
, snd(snd(snd(msgval(e))))>)) ~ pv11_p1_p2a'send(C\000Cmd;f) loc(e)
<fst(msgval(e))
, fst(snd(msgval(e)))
, fst(snd(snd(msgval(e))))
, snd(snd(snd(msgval(e))))>⌉ (-1)⋅
THENA (RepUR ``pv11_p1_p2a'send mk-msg-interface make-Msg mk-msg`` 0 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. Inj(Id;ℤ;ldrs_uid)@i
19. e : E@i
20. ∀e1:E
((e1 < e)
⇒ (∀b:pv11_p1_Ballot_Num(). ∀k:ℤ. ∀v:Cmd.
(pv11_p1_pvalue{i:l}(Cmd;ldrs_uid;accpts;es;e1;<b, k, v>;f)
⇒ (∀e':E. ∀w:Cmd.
(pv11_p1_decision{i:l}(Cmd;f;accpts;ldrs;ldrs_uid;reps;es;e';k;w)
⇒ has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) (snd(snd(snd(msgval(e'))))) b))
⇒ (w = v ∈ Cmd))))))
21. b : pv11_p1_Ballot_Num()@i
22. k : ℤ@i
23. v : Cmd@i
24. pv11_p1_pvalue{i:l}(Cmd;ldrs_uid;accpts;es;e;<b, k, v>;f)@i'
25. header(e) = ``pv11_p1 p2a`` ∈ Name
26. <b, k, v> = (snd(msgval(e))) ∈ (pv11_p1_Ballot_Num() × ℤ × Cmd)
27. e' : E@i
28. w : Cmd@i
29. pv11_p1_decision{i:l}(Cmd;f;accpts;ldrs;ldrs_uid;reps;es;e';k;w)@i'
30. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())@i
31. ↑(pv11_p1_leq_bnum(ldrs_uid) (snd(snd(snd(msgval(e'))))) b)@i
32. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
33. ∀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'))))
34. e1 : E
35. delay : ℤ
36. (e1 < e)
37. pv11_p1_p2a'send(Cmd;f) loc(e)
<fst(msgval(e)), fst(snd(msgval(e))), fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> ∈
pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e1)@i
⊢ w = v ∈ 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. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i
19. e : E@i
20. \mforall{}e1:E
((e1 < e)
{}\mRightarrow{} (\mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}k:\mBbbZ{}. \mforall{}v:Cmd.
(pv11\_p1\_pvalue\{i:l\}(Cmd;ldrs$_{uid}$;accpts;es;e1;<b, k, v>f)
{}\mRightarrow{} (\mforall{}e':E. \mforall{}w:Cmd.
(pv11\_p1\_decision\{i:l\}(Cmd;f;accpts;ldrs;ldrs$_{uid}$;reps;es;\000Ce';k;w)
{}\mRightarrow{} has-es-info-type(es;e';f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num())
{}\mRightarrow{} (\muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) (snd(snd(snd(msgval(e')))\000C)) b))
{}\mRightarrow{} (w = v))))))
21. b : pv11\_p1\_Ballot\_Num()@i
22. k : \mBbbZ{}@i
23. v : Cmd@i
24. pv11\_p1\_pvalue\{i:l\}(Cmd;ldrs$_{uid}$;accpts;es;e;<b, k, v>f)@i'
25. header(e) = ``pv11\_p1 p2a``
26. <b, k, v> = (snd(msgval(e)))
27. e' : E@i
28. w : Cmd@i
29. pv11\_p1\_decision\{i:l\}(Cmd;f;accpts;ldrs;ldrs$_{uid}$;reps;es;e';k;w)@i'
30. has-es-info-type(es;e';f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num())@i
31. \muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) (snd(snd(snd(msgval(e'))))) b)@i
32. has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd)
33. \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'))))
34. e1 : E
35. delay : \mBbbZ{}
36. (e1 < e)
37. make-msg-interface(delay;loc(e);mk-msg(es-info-auth(es;e);``pv11\_p1 p2a``;<fst(msgval(e))
, fst(snd(msgval(e)))
, fst(snd(snd(...)))
, snd(snd(snd(...)))>)\000C) \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid}$;reps;f)(e1)@i
\mvdash{} w = v
By
Latex:
((Subst \mkleeneopen{}delay \msim{} 0\mkleeneclose{} (-1)\mcdot{} THENA (RewriteWithILF (-1) THEN Auto))
THEN (Subst \mkleeneopen{}es-info-auth(es;e) \msim{} ff\mkleeneclose{} (-1)\mcdot{} THENA (RewriteWithILF (-1) THEN Auto))
THEN (Subst \mkleeneopen{}make-msg-interface(0;loc(e);mk-msg(ff;``pv11\_p1 p2a``;<fst(msgval(e))
, fst(snd(msgval(e)))
, fst(snd(snd(msgval(e))))
, snd(snd(snd(msgval(e))))>)) \msim{} \000Cpv11\_p1\_p2a'send(Cmd;f) loc(e)
<fst(msgval(e))
, fst(snd(msgval(e)))
, fst(snd(snd(msgval(e))))
, snd(snd(snd(msgval(e))))>\mkleeneclose{} (-1)\mcdot{}
THENA (RepUR ``pv11\_p1\_p2a'send mk-msg-interface make-Msg mk-msg`` 0 THEN Auto)
))
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