Step
*
2
2
of Lemma
pv11_p1_ldr_mem_adopted
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. e1 : E@i
14. e2 : E@i
15. ldrs_uid : Id ─→ ℤ@i
16. v5 : pv11_p1_Ballot_Num()@i
17. v9 : 𝔹@i
18. v10 : (ℤ × Cmd) List@i
19. v3 : pv11_p1_Ballot_Num()@i
20. v7 : 𝔹@i
21. v8 : (ℤ × Cmd) List@i
22. x4 : ℤ@i
23. x5 : Cmd@i
24. (e1 <loc e2)@i
25. inl <x4, x5> ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e1)@i
26. <v5, v9, v10> ∈
Memory-loc-class(pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + ...;λloc.{pv11_p1_init_leader(Cmd)
loc};... (+) ...)(e1)
27. <v3, v7, v8> ∈
Memory-loc-class(pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + ...;λloc.{pv11_p1_init_leader(Cmd)
loc};... (+) ...)(e2)
28. x6 : pv11_p1_Ballot_Num()@i
29. x7 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
30. x2 : ℤ@i
31. x3 : Cmd@i
32. s5 : pv11_p1_Ballot_Num()@i
33. s9 : 𝔹@i
34. s10 : (ℤ × Cmd) List@i
35. s3 : pv11_p1_Ballot_Num()@i
36. s7 : 𝔹@i
37. s8 : (ℤ × Cmd) List@i
38. e : E@i
39. e' : E@i
40. e1 ≤loc e @i
41. (e <loc e')@i
42. (e' <loc e2)@i
43. inr (inl <x6, x7>) ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(\000Ce)@i
44. <s5, s9, s10> ∈
Memory-loc-class(pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + ...;λloc.{pv11_p1_init_leader(Cmd)
loc};... (+) ...)(e)@i
45. inl <x2, x3> ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e')@i
46. <s3, s7, s8> ∈
Memory-loc-class(pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + ...;λloc.{pv11_p1_init_leader(Cmd)
loc};... (+) ...)(e')@i
47. s5 = x6 ∈ pv11_p1_Ballot_Num()@i
48. s : ℤ@i
49. b : pv11_p1_Ballot_Num()@i
50. c : Cmd@i
51. (<b, s, c> ∈ x7)@i
52. ∀s:ℤ. ∀b:pv11_p1_Ballot_Num(). ∀c:Cmd. (((<s, c> ∈ s10) ∨ (<b, s, c> ∈ x7)) ⇒ (↑(pv11_p1_in_domain(Cmd) s s8)))@i
53. ∀b:pv11_p1_Ballot_Num(). ∀c:Cmd. (((<s, c> ∈ s10) ∨ (<b, s, c> ∈ x7)) ⇒ (↑(pv11_p1_in_domain(Cmd) s s8)))
54. ∀b:pv11_p1_Ballot_Num(). ∀c:Cmd. (((<x2, c> ∈ s10) ∨ (<b, x2, c> ∈ x7)) ⇒ (↑(pv11_p1_in_domain(Cmd) x2 s8)))
⊢ ∃y:ℤ × Cmd. ((y ∈ if x2 ∈b map(λx.(fst(x));s8)) then s8 else pv11_p1_add_if_new() pv11_p1_same_proposal(Cmd) <x2, x3> \000Cs8 fi ) ∧ (s = (fst(y)) ∈ ℤ))
BY
{ ((InstHyp [⌈b⌉;⌈c⌉] (-2)⋅ THENA Auto)
THEN RepUR ``pv11_p1_in_domain`` (-1)
THEN (RW assert_pushdownC (-1) THENA Auto)
THEN (RW ListC (-1) THENA Auto)
THEN AllReduce
THEN ExRepD
THEN AutoSplit
THEN (InstConcl [⌈y⌉]⋅ THEN Auto)
THEN BLemma `pv11_p1_add_if_new_iff2`
THEN Auto) }
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. e1 : E@i
14. e2 : E@i
15. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i
16. v5 : pv11\_p1\_Ballot\_Num()@i
17. v9 : \mBbbB{}@i
18. v10 : (\mBbbZ{} \mtimes{} Cmd) List@i
19. v3 : pv11\_p1\_Ballot\_Num()@i
20. v7 : \mBbbB{}@i
21. v8 : (\mBbbZ{} \mtimes{} Cmd) List@i
22. x4 : \mBbbZ{}@i
23. x5 : Cmd@i
24. (e1 <loc e2)@i
25. inl <x4, x5> \mmember{}
pv11\_p1\_propose'base(Cmd;f) (+) pv11\_p1\_adopted'base(Cmd;f) (+) pv11\_p1\_preempted'base(Cmd;f)(
e1)@i
26. <v5, v9, v10> \mmember{}
Memory-loc-class(pv11\_p1\_on\_propose(Cmd) + pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid}\000C$) + ...;...;...)(e1)
27. <v3, v7, v8> \mmember{}
Memory-loc-class(pv11\_p1\_on\_propose(Cmd) + pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid}\000C$) + ...;...;...)(e2)
28. x6 : pv11\_p1\_Ballot\_Num()@i
29. x7 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
30. x2 : \mBbbZ{}@i
31. x3 : Cmd@i
32. s5 : pv11\_p1\_Ballot\_Num()@i
33. s9 : \mBbbB{}@i
34. s10 : (\mBbbZ{} \mtimes{} Cmd) List@i
35. s3 : pv11\_p1\_Ballot\_Num()@i
36. s7 : \mBbbB{}@i
37. s8 : (\mBbbZ{} \mtimes{} Cmd) List@i
38. e : E@i
39. e' : E@i
40. e1 \mleq{}loc e @i
41. (e <loc e')@i
42. (e' <loc e2)@i
43. inr (inl <x6, x7>) \mmember{}
pv11\_p1\_propose'base(Cmd;f) (+) pv11\_p1\_adopted'base(Cmd;f) (+) pv11\_p1\_preempted'base(Cmd;f)(
e)@i
44. <s5, s9, s10> \mmember{}
Memory-loc-class(pv11\_p1\_on\_propose(Cmd) + pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid}\000C$) + ...;...;...)(
e)@i
45. inl <x2, x3> \mmember{}
pv11\_p1\_propose'base(Cmd;f) (+) pv11\_p1\_adopted'base(Cmd;f) (+) pv11\_p1\_preempted'base(Cmd;f)(
e')@i
46. <s3, s7, s8> \mmember{}
Memory-loc-class(pv11\_p1\_on\_propose(Cmd) + pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid}\000C$) + ...;...;...)(
e')@i
47. s5 = x6@i
48. s : \mBbbZ{}@i
49. b : pv11\_p1\_Ballot\_Num()@i
50. c : Cmd@i
51. (<b, s, c> \mmember{} x7)@i
52. \mforall{}s:\mBbbZ{}. \mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}c:Cmd.
(((<s, c> \mmember{} s10) \mvee{} (<b, s, c> \mmember{} x7)) {}\mRightarrow{} (\muparrow{}(pv11\_p1\_in\_domain(Cmd) s s8)))@i
53. \mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}c:Cmd.
(((<s, c> \mmember{} s10) \mvee{} (<b, s, c> \mmember{} x7)) {}\mRightarrow{} (\muparrow{}(pv11\_p1\_in\_domain(Cmd) s s8)))
54. \mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}c:Cmd.
(((<x2, c> \mmember{} s10) \mvee{} (<b, x2, c> \mmember{} x7)) {}\mRightarrow{} (\muparrow{}(pv11\_p1\_in\_domain(Cmd) x2 s8)))
\mvdash{} \mexists{}y:\mBbbZ{} \mtimes{} Cmd
((y \mmember{} if x2 \mmember{}\msubb{} map(\mlambda{}x.(fst(x));s8))
then s8
else pv11\_p1\_add\_if\_new() pv11\_p1\_same\_proposal(Cmd) <x2, x3> s8
fi )
\mwedge{} (s = (fst(y))))
By
Latex:
((InstHyp [\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{}] (-2)\mcdot{} THENA Auto)
THEN RepUR ``pv11\_p1\_in\_domain`` (-1)
THEN (RW assert\_pushdownC (-1) THENA Auto)
THEN (RW ListC (-1) THENA Auto)
THEN AllReduce
THEN ExRepD
THEN AutoSplit
THEN (InstConcl [\mkleeneopen{}y\mkleeneclose{}]\mcdot{} THEN Auto)
THEN BLemma `pv11\_p1\_add\_if\_new\_iff2`
THEN Auto)
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