Step
*
9
of Lemma
pv11_p1_ldr_mem_propose
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. y : pv11_p1_Ballot_Num()@i
23. (e1 <loc e2)@i
24. inr inr y ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e1)@i
25. <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)
26. <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)
27. x2 : ℤ@i
28. x3 : Cmd@i
29. x4 : pv11_p1_Ballot_Num()@i
30. x5 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
31. s5 : pv11_p1_Ballot_Num()@i
32. s9 : 𝔹@i
33. s10 : (ℤ × Cmd) List@i
34. s3 : pv11_p1_Ballot_Num()@i
35. s7 : 𝔹@i
36. s8 : (ℤ × Cmd) List@i
37. e : E@i
38. e' : E@i
39. e1 ≤loc e @i
40. (e <loc e')@i
41. (e' <loc e2)@i
42. inl <x2, x3> ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e)@i
43. <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
44. inr (inl <x4, x5>) ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(\000Ce')@i
45. <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
46. x4 = s3 ∈ pv11_p1_Ballot_Num()
47. ¬(x2 ∈ map(λx.(fst(x));s10))
48. y2 : ℤ
49. y3 : Cmd
50. (<y2, y3> ∈ s8)
51. x2 = y2 ∈ ℤ
⊢ ∃y:ℤ × Cmd. ((y ∈ pv11_p1_update_proposals(Cmd) s8 (pv11_p1_pmax(Cmd;ldrs_uid) x5)) ∧ (x2 = (fst(y)) ∈ ℤ))
BY
{ ((InstLemma `pv11_p1_upd_left` [⌈Cmd⌉;⌈s8⌉;⌈pv11_p1_pmax(Cmd;ldrs_uid) x5⌉;⌈y2⌉;⌈y3⌉]⋅ THENA Auto)
THEN ExRepD
THEN InstConcl [⌈<y2, 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. y : pv11\_p1\_Ballot\_Num()@i
23. (e1 <loc e2)@i
24. inr inr y \mmember{}
pv11\_p1\_propose'base(Cmd;f) (+) pv11\_p1\_adopted'base(Cmd;f) (+) pv11\_p1\_preempted'base(Cmd;f)(
e1)@i
25. <v5, v9, v10> \mmember{}
Memory-loc-class(pv11\_p1\_on\_propose(Cmd) + pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid}\000C$) + ...;...;...)(e1)
26. <v3, v7, v8> \mmember{}
Memory-loc-class(pv11\_p1\_on\_propose(Cmd) + pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid}\000C$) + ...;...;...)(e2)
27. x2 : \mBbbZ{}@i
28. x3 : Cmd@i
29. x4 : pv11\_p1\_Ballot\_Num()@i
30. x5 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
31. s5 : pv11\_p1\_Ballot\_Num()@i
32. s9 : \mBbbB{}@i
33. s10 : (\mBbbZ{} \mtimes{} Cmd) List@i
34. s3 : pv11\_p1\_Ballot\_Num()@i
35. s7 : \mBbbB{}@i
36. s8 : (\mBbbZ{} \mtimes{} Cmd) List@i
37. e : E@i
38. e' : E@i
39. e1 \mleq{}loc e @i
40. (e <loc e')@i
41. (e' <loc e2)@i
42. inl <x2, x3> \mmember{}
pv11\_p1\_propose'base(Cmd;f) (+) pv11\_p1\_adopted'base(Cmd;f) (+) pv11\_p1\_preempted'base(Cmd;f)(
e)@i
43. <s5, s9, s10> \mmember{}
Memory-loc-class(pv11\_p1\_on\_propose(Cmd) + pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid}\000C$) + ...;...;...)(
e)@i
44. inr (inl <x4, x5>) \mmember{}
pv11\_p1\_propose'base(Cmd;f) (+) pv11\_p1\_adopted'base(Cmd;f) (+) pv11\_p1\_preempted'base(Cmd;f)(
e')@i
45. <s3, s7, s8> \mmember{}
Memory-loc-class(pv11\_p1\_on\_propose(Cmd) + pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid}\000C$) + ...;...;...)(
e')@i
46. x4 = s3
47. \mneg{}(x2 \mmember{} map(\mlambda{}x.(fst(x));s10))
48. y2 : \mBbbZ{}
49. y3 : Cmd
50. (<y2, y3> \mmember{} s8)
51. x2 = y2
\mvdash{} \mexists{}y:\mBbbZ{} \mtimes{} Cmd
((y \mmember{} pv11\_p1\_update\_proposals(Cmd) s8 (pv11\_p1\_pmax(Cmd;ldrs$_{uid}$) x5)) \mwedge{}\000C (x2 = (fst(y))))
By
Latex:
((InstLemma `pv11\_p1\_upd\_left` [\mkleeneopen{}Cmd\mkleeneclose{};\mkleeneopen{}s8\mkleeneclose{};\mkleeneopen{}pv11\_p1\_pmax(Cmd;ldrs$_{uid}$) x5\mkleeneclose{};\mkleeneopen{}\000Cy2\mkleeneclose{};\mkleeneopen{}y3\mkleeneclose{}]\mcdot{} THENA Auto)
THEN ExRepD
THEN InstConcl [\mkleeneopen{}<y2, c'>\mkleeneclose{}]\mcdot{}
THEN Auto)
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