Step
*
1
2
1
1
1
of Lemma
pv11_p1_append_news_iff
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∈L1 @ [<u1, u3, u4> / v].∃e:E. (x ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e))))@i
29. <p1, p3, p4> = <u1, u3, u4> ∈ (pv11_p1_Ballot_Num() × ℤ × Cmd)@i
30. h : (∀x∈(pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) <u1, u3, u4> L1) @ v.∃e:E. (x ∈ snd(pv11_p1_AcceptorStateFun(\000CCmd;ldrs_uid;f;es;e))))
31. (<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:
pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) <u1, u3, u4> L1))
⇒ ((<p1, p3, p4> ∈ pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) <u1, u3, u4> L1) ∨ (<p1, p3, p4> ∈ v))
32. (<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:
pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) <u1, u3, u4> L1)) ⇐ (<p1, p3, p4> ∈ pv11_p1_add_if_n\000Cew() pv11_p1_same_pvalue(Cmd) <u1, u3, u4> L1) ∨ (<p1, p3, p4> ∈ v)
33. (∃z∈L1. ↑(pv11_p1_same_pvalue(Cmd) <u1, u3, u4> z))
⊢ ((<p1, p3, p4> ∈ L1) ∨ False) ∨ (<p1, p3, p4> ∈ v)
BY
{ ((RWO "l_all_iff" (-4) THENA Auto)
THEN (RWO "-5" 0 THENA Auto)
THEN (RWO "l_all_iff" (-6) THENA Auto)
THEN (InstHyp [⌈<u1, u3, u4>⌉] (-6)⋅ THENA Auto)
THEN ExRepD
THEN (RWO "l_exists_iff" (-3) THENA Auto)
THEN ExRepD
THEN (InstHyp [⌈z⌉] (-8)⋅
THENA (Auto
THEN (RW ListC 0 THEN Auto)
THEN (OrLeft THENA Auto)
THEN BLemma `pv11_p1_add_if_new_iff2`
THEN Auto)
)
THEN ExRepD
THEN DProdsAndUnions) }
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)@i
30. ∀x:pv11_p1_Ballot_Num() × ℤ × Cmd
((x ∈ (pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) <u1, u3, u4> L1) @ v) ⇒ (∃e:E. (x ∈ snd(pv11_p1_AcceptorStat\000CeFun(Cmd;ldrs_uid;f;es;e)))))
31. (<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:
pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) <u1, u3, u4> L1))
⇒ ((<p1, p3, p4> ∈ pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) <u1, u3, u4> L1) ∨ (<p1, p3, p4> ∈ v))
32. (<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:
pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) <u1, u3, u4> L1)) ⇐ (<p1, p3, p4> ∈ pv11_p1_add_if_n\000Cew() pv11_p1_same_pvalue(Cmd) <u1, u3, u4> L1) ∨ (<p1, p3, p4> ∈ v)
33. z1 : pv11_p1_Ballot_Num()
34. z3 : ℤ
35. z4 : Cmd
36. (<z1, z3, z4> ∈ L1)
37. ↑(pv11_p1_same_pvalue(Cmd) <u1, u3, u4> <z1, z3, z4>)
38. e : E
39. (<u1, u3, u4> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)))
40. e1 : E
41. (<z1, z3, z4> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e1)))
⊢ ((<u1, u3, u4> ∈ L1) ∨ False) ∨ (<u1, u3, u4> ∈ v)
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. p1 : pv11\_p1\_Ballot\_Num()@i
18. p3 : \mBbbZ{}@i
19. p4 : Cmd@i
20. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i
21. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i
22. u1 : pv11\_p1\_Ballot\_Num()@i
23. u3 : \mBbbZ{}@i
24. u4 : Cmd@i
25. v : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
26. \mforall{}L1:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List
((\mforall{}x\mmember{}L1 @ v.\mexists{}e:E. (x \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;\000Ce))))
{}\mRightarrow{} ((<p1, p3, p4> \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{} (<p1, p3, p4> \mmember{} L1) \mvee{} (<p1, p3, p4> \mmember{} v)))@i
27. L1 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
28. (\mforall{}x\mmember{}L1 @ [<u1, u3, u4> / v].\mexists{}e:E. (x \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid\mbackslash{}f\000Cf7d$;f;es;e))))@i
29. <p1, p3, p4> = <u1, u3, u4>@i
30. h : (\mforall{}x\mmember{}(pv11\_p1\_add\_if\_new() pv11\_p1\_same\_pvalue(Cmd) <u1, u3, u4> L1) @ v.\mexists{}e:E. (x \mmember{} snd(pv11\_\000Cp1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e))))
31. (<p1, p3, p4> \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:
pv11\_p1\_add\_if\_new() pv11\_p1\_same\_pvalue(Cmd) <u1, u3, u4> L1))
{}\mRightarrow{} ((<p1, p3, p4> \mmember{} pv11\_p1\_add\_if\_new() pv11\_p1\_same\_pvalue(Cmd) <u1, u3, u4> L1) \mvee{} (<p1, p3, p4> \mmember{}\000C v))
32. (<p1, p3, p4> \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:
pv11\_p1\_add\_if\_new() pv11\_p1\_same\_pvalue(Cmd) <u1, u3, u4> L1)) \mLeftarrow{}{} (<p1, p3, p4\000C> \mmember{} pv11\_p1\_add\_if\_new()
pv11\_p1\_same\_pvalue(Cmd)
<u1, u3, u4>
L1)
\mvee{} (<p1, p3, p4> \mmember{} v)
33. (\mexists{}z\mmember{}L1. \muparrow{}(pv11\_p1\_same\_pvalue(Cmd) <u1, u3, u4> z))
\mvdash{} ((<p1, p3, p4> \mmember{} L1) \mvee{} False) \mvee{} (<p1, p3, p4> \mmember{} v)
By
Latex:
((RWO "l\_all\_iff" (-4) THENA Auto)
THEN (RWO "-5" 0 THENA Auto)
THEN (RWO "l\_all\_iff" (-6) THENA Auto)
THEN (InstHyp [\mkleeneopen{}<u1, u3, u4>\mkleeneclose{}] (-6)\mcdot{} THENA Auto)
THEN ExRepD
THEN (RWO "l\_exists\_iff" (-3) THENA Auto)
THEN ExRepD
THEN (InstHyp [\mkleeneopen{}z\mkleeneclose{}] (-8)\mcdot{}
THENA (Auto
THEN (RW ListC 0 THEN Auto)
THEN (OrLeft THENA Auto)
THEN BLemma `pv11\_p1\_add\_if\_new\_iff2`
THEN Auto)
)
THEN ExRepD
THEN DProdsAndUnions)
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