Step
*
1
7
2
of Lemma
pv11_p1_validity-lemma
.....assertion.....
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(). ∀start:E. (start ≤loc e ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e;f))
23. (header(e) = ``pv11_p1 adopted`` ∈ Name)
⇒ has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
⇒ pv11_p1_valid-adopted-message(Cmd;es;e;f)
24. (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))
⇒ pv11_p1_valid-p1b-message(Cmd;es;e;f)
25. header(e) = ``pv11_p1 p2a`` ∈ Name@i
26. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)@i
27. ∀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'))))
28. e' : E
29. delay : ℤ
30. (e' < e)
31. loc(e') ↓∈ ldrs
32. loc(e) ↓∈ accpts
33. delay = 0 ∈ ℤ
34. es-info-auth(es;e) = pv11_p1_init_active()
35. (fst(msgval(e))) = loc(e') ∈ Id
36. header(e') = ``pv11_p1 adopted`` ∈ Name
37. has-es-info-type(es;e';f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
38. (fst(msgval(e'))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num()
39. v2 : pv11_p1_Ballot_Num()
40. <v2, fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> ↓∈ snd(msgval(e'))
41. ¬(∃z5:pv11_p1_Ballot_Num(). ∃z8:Cmd. ((↑(v2 < z5)) ∧ (<z5, fst(snd(snd(msgval(e)))), z8> ∈ snd(msgval(e')))))
42. (fst(snd(msgval(e)))) = (fst(msgval(e'))) ∈ pv11_p1_Ballot_Num()
43. (<v2, fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> ∈ snd(msgval(e')))
44. e' c≤ e
⊢ pv11_p1_valid-proposal(Cmd;es;e';<fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))>;f)
BY
{ ((Assert pv11_p1_valid-adopted-message(Cmd;es;e';f) BY
(BackThruSomeHyp THEN Auto))
THEN RepeatFor 2 (UnfoldTopAb (-1))
THEN (InstLemma `l_all_iff` [⌈ℤ × Cmd⌉]⋅ THENA Auto)
THEN (RWO "-1" (-2) THENA Auto)
THEN Thin (-1)) }
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
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(). ∀start:E. (start ≤loc e ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e;f))
23. (header(e) = ``pv11_p1 adopted`` ∈ Name)
⇒ has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
⇒ pv11_p1_valid-adopted-message(Cmd;es;e;f)
24. (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))
⇒ pv11_p1_valid-p1b-message(Cmd;es;e;f)
25. header(e) = ``pv11_p1 p2a`` ∈ Name@i
26. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)@i
27. ∀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'))))
28. e' : E
29. delay : ℤ
30. (e' < e)
31. loc(e') ↓∈ ldrs
32. loc(e) ↓∈ accpts
33. delay = 0 ∈ ℤ
34. es-info-auth(es;e) = pv11_p1_init_active()
35. (fst(msgval(e))) = loc(e') ∈ Id
36. header(e') = ``pv11_p1 adopted`` ∈ Name
37. has-es-info-type(es;e';f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
38. (fst(msgval(e'))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num()
39. v2 : pv11_p1_Ballot_Num()
40. <v2, fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> ↓∈ snd(msgval(e'))
41. ¬(∃z5:pv11_p1_Ballot_Num(). ∃z8:Cmd. ((↑(v2 < z5)) ∧ (<z5, fst(snd(snd(msgval(e)))), z8> ∈ snd(msgval(e')))))
42. (fst(snd(msgval(e)))) = (fst(msgval(e'))) ∈ pv11_p1_Ballot_Num()
43. (<v2, fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> ∈ snd(msgval(e')))
44. e' c≤ e
45. ∀p:ℤ × Cmd. ((p ∈ map(λp.(snd(p));snd(msgval(e')))) ⇒ pv11_p1_valid-proposal(Cmd;es;e';p;f))
⊢ pv11_p1_valid-proposal(Cmd;es;e';<fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))>;f)
Latex:
Latex:
.....assertion.....
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. \mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}start:E.
(start \mleq{}loc e {}\mRightarrow{} pv11\_p1\_valid-ScoutState(Cmd;accpts;b;start;es;e;f))
23. (header(e) = ``pv11\_p1 adopted``)
{}\mRightarrow{} has-es-info-type(es;e;f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))
{}\mRightarrow{} pv11\_p1\_valid-adopted-message(Cmd;es;e;f)
24. (header(e) = ``pv11\_p1 p1b``)
{}\mRightarrow{} 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))
{}\mRightarrow{} pv11\_p1\_valid-p1b-message(Cmd;es;e;f)
25. header(e) = ``pv11\_p1 p2a``@i
26. has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd)@i
27. \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'))))
28. e' : E
29. delay : \mBbbZ{}
30. (e' < e)
31. loc(e') \mdownarrow{}\mmember{} ldrs
32. loc(e) \mdownarrow{}\mmember{} accpts
33. delay = 0
34. es-info-auth(es;e) = pv11\_p1\_init\_active()
35. (fst(msgval(e))) = loc(e')
36. header(e') = ``pv11\_p1 adopted``
37. has-es-info-type(es;e';f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))
38. (fst(msgval(e'))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e')))
39. v2 : pv11\_p1\_Ballot\_Num()
40. <v2, fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> \mdownarrow{}\mmember{} snd(msgval(e'))
41. \mneg{}(\mexists{}z5:pv11\_p1\_Ballot\_Num()
\mexists{}z8:Cmd. ((\muparrow{}(v2 < z5)) \mwedge{} (<z5, fst(snd(snd(msgval(e)))), z8> \mmember{} snd(msgval(e')))))
42. (fst(snd(msgval(e)))) = (fst(msgval(e')))
43. (<v2, fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> \mmember{} snd(msgval(e')))
44. e' c\mleq{} e
\mvdash{} pv11\_p1\_valid-proposal(Cmd;es;e';<fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))>f)
By
Latex:
((Assert pv11\_p1\_valid-adopted-message(Cmd;es;e';f) BY
(BackThruSomeHyp THEN Auto))
THEN RepeatFor 2 (UnfoldTopAb (-1))
THEN (InstLemma `l\_all\_iff` [\mkleeneopen{}\mBbbZ{} \mtimes{} Cmd\mkleeneclose{}]\mcdot{} THENA Auto)
THEN (RWO "-1" (-2) THENA Auto)
THEN Thin (-1))
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