Step
*
1
2
2
2
1
of Lemma
pv11_p1_append_news_iff2
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. pv : pv11_p1_Ballot_Num() × ℤ × Cmd@i
18. Inj(Id;ℤ;ldrs_uid)@i
19. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
20. u : pv11_p1_Ballot_Num() × ℤ × Cmd@i
21. v : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
22. ∀L1:(pv11_p1_Ballot_Num() × ℤ × Cmd) List
((∀x∈L1 @ v.↓∃e:E. (x ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e))))
⇒ ((pv ∈ accumulate (with value a and list item x):
pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) x a
over list:
v
with starting value:
L1))
⇐⇒ (pv ∈ L1) ∨ (pv ∈ v)))@i
23. L1 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
24. ∀x:pv11_p1_Ballot_Num() × ℤ × Cmd
((x ∈ L1 @ [u / v]) ⇒ (↓∃e:E. (x ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)))))
25. pv = u ∈ (pv11_p1_Ballot_Num() × ℤ × Cmd)
26. z : pv11_p1_Ballot_Num() × ℤ × Cmd
27. (z ∈ L1)
28. ↑(pv11_p1_same_pvalue(Cmd) u z)
29. ↓∃e:E. (u ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)))
30. ↓∃e:E. (z ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)))
⊢ (pv ∈ L1)
BY
{ (DProdsAndUnions
THEN RepeatFor 2 ((MemorySimpleAssertReasoningOne (-3) THEN Auto))
THEN MemorySimpleAssertReasoningOne (-4)
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. p1 : pv11_p1_Ballot_Num()@i
18. p3 : ℤ@i
19. p4 : Cmd@i
20. Inj(Id;ℤ;ldrs_uid)@i
21. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
22. u1 : pv11_p1_Ballot_Num()@i
23. u3 : ℤ@i
24. u4 : Cmd@i
25. v : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
26. ∀L1:(pv11_p1_Ballot_Num() × ℤ × Cmd) List
((∀x∈L1 @ v.↓∃e:E. (x ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e))))
⇒ ((<p1, p3, p4> ∈ accumulate (with value a and list item x):
pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) x a
over list:
v
with starting value:
L1))
⇐⇒ (<p1, p3, p4> ∈ L1) ∨ (<p1, p3, p4> ∈ v)))@i
27. L1 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
28. ∀x:pv11_p1_Ballot_Num() × ℤ × Cmd
((x ∈ L1 @ [<u1, u3, u4> / v]) ⇒ (↓∃e:E. (x ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)))))
29. <p1, p3, p4> = <u1, u3, u4> ∈ (pv11_p1_Ballot_Num() × ℤ × Cmd)
30. z1 : pv11_p1_Ballot_Num()
31. z3 : ℤ
32. z4 : Cmd
33. (<z1, z3, z4> ∈ L1)
34. u1 = z1 ∈ (ℤ × Id?)
35. u3 = z3 ∈ ℤ
36. ↓∃e:E. (<u1, u3, u4> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)))
37. ↓∃e:E. (<z1, z3, z4> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)))
⊢ (<p1, p3, p4> ∈ L1)
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. pv : pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd@i
18. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i
19. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i
20. u : pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd@i
21. v : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
22. \mforall{}L1:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List
((\mforall{}x\mmember{}L1 @ v.\mdownarrow{}\mexists{}e:E. (x \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es\000C;e))))
{}\mRightarrow{} ((pv \mmember{} accumulate (with value a and list item x):
pv11\_p1\_add\_if\_new() pv11\_p1\_same\_pvalue(Cmd) x a
over list:
v
with starting value:
L1))
\mLeftarrow{}{}\mRightarrow{} (pv \mmember{} L1) \mvee{} (pv \mmember{} v)))@i
23. L1 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
24. \mforall{}x:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd
((x \mmember{} L1 @ [u / v]) {}\mRightarrow{} (\mdownarrow{}\mexists{}e:E. (x \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid\mbackslash{}ff\000C7d$;f;es;e)))))
25. pv = u
26. z : pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd
27. (z \mmember{} L1)
28. \muparrow{}(pv11\_p1\_same\_pvalue(Cmd) u z)
29. \mdownarrow{}\mexists{}e:E. (u \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e)))
30. \mdownarrow{}\mexists{}e:E. (z \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e)))
\mvdash{} (pv \mmember{} L1)
By
Latex:
(DProdsAndUnions
THEN RepeatFor 2 ((MemorySimpleAssertReasoningOne (-3) THEN Auto))
THEN MemorySimpleAssertReasoningOne (-4)
THEN Auto)
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