Step
*
2
8
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. x3 : ℤ@i
23. x4 : Cmd@i
24. (e1 <loc e2)@i
25. inl <x3, x4> ∈ 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. x7 : pv11_p1_Ballot_Num()@i
29. x8 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
30. x5 : pv11_p1_Ballot_Num()@i
31. x6 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@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 <x7, x8>) ∈ 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. inr (inl <x5, x6>) ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(\000Ce')@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. x5 = s3 ∈ pv11_p1_Ballot_Num()
48. s5 = x7 ∈ pv11_p1_Ballot_Num()@i
49. s : ℤ@i
50. b : pv11_p1_Ballot_Num()@i
51. c : Cmd@i
52. (<b, s, c> ∈ x8)@i
53. ∀s:ℤ. ∀b:pv11_p1_Ballot_Num(). ∀c:Cmd. (((<s, c> ∈ s10) ∨ (<b, s, c> ∈ x8)) ⇒ (↑(pv11_p1_in_domain(Cmd) s s8)))@i
54. ∀b:pv11_p1_Ballot_Num(). ∀c:Cmd. (((<s, c> ∈ s10) ∨ (<b, s, c> ∈ x8)) ⇒ (↑(pv11_p1_in_domain(Cmd) s s8)))
⊢ ∃y:ℤ × Cmd. ((y ∈ pv11_p1_update_proposals(Cmd) s8 (pv11_p1_pmax(Cmd;ldrs_uid) x6)) ∧ (s = (fst(y)) ∈ ℤ))
BY
{ ((InstHyp [⌈b⌉;⌈c⌉] (-1)⋅ 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 (DProdsAndUnions THEN AllReduce)
THEN (InstLemma `pv11_p1_upd_left` [⌈Cmd⌉;⌈s8⌉;⌈pv11_p1_pmax(Cmd;ldrs_uid) x6⌉;⌈s⌉;⌈y2⌉]⋅ THENA Auto)
THEN ExRepD
THEN InstConcl [⌈<s, c'>⌉]⋅
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. x3 : \mBbbZ{}@i
23. x4 : Cmd@i
24. (e1 <loc e2)@i
25. inl <x3, x4> \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. x7 : pv11\_p1\_Ballot\_Num()@i
29. x8 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
30. x5 : pv11\_p1\_Ballot\_Num()@i
31. x6 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@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 <x7, x8>) \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. inr (inl <x5, x6>) \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. x5 = s3
48. s5 = x7@i
49. s : \mBbbZ{}@i
50. b : pv11\_p1\_Ballot\_Num()@i
51. c : Cmd@i
52. (<b, s, c> \mmember{} x8)@i
53. \mforall{}s:\mBbbZ{}. \mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}c:Cmd.
(((<s, c> \mmember{} s10) \mvee{} (<b, s, c> \mmember{} x8)) {}\mRightarrow{} (\muparrow{}(pv11\_p1\_in\_domain(Cmd) s s8)))@i
54. \mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}c:Cmd.
(((<s, c> \mmember{} s10) \mvee{} (<b, s, c> \mmember{} x8)) {}\mRightarrow{} (\muparrow{}(pv11\_p1\_in\_domain(Cmd) s s8)))
\mvdash{} \mexists{}y:\mBbbZ{} \mtimes{} Cmd
((y \mmember{} pv11\_p1\_update\_proposals(Cmd) s8 (pv11\_p1\_pmax(Cmd;ldrs$_{uid}$) x6)) \mwedge{}\000C (s = (fst(y))))
By
Latex:
((InstHyp [\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{}] (-1)\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 (DProdsAndUnions THEN AllReduce)
THEN (InstLemma `pv11\_p1\_upd\_left` [\mkleeneopen{}Cmd\mkleeneclose{};\mkleeneopen{}s8\mkleeneclose{};\mkleeneopen{}pv11\_p1\_pmax(Cmd;ldrs$_{uid}$) \000Cx6\mkleeneclose{};\mkleeneopen{}s\mkleeneclose{};\mkleeneopen{}y2\mkleeneclose{}]\mcdot{}
THENA Auto
)
THEN ExRepD
THEN InstConcl [\mkleeneopen{}<s, c'>\mkleeneclose{}]\mcdot{}
THEN Auto)
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