Step
*
of Lemma
pv11_p1_ldr_mem_propose
∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀e1,e2:E. ∀ldrs_uid:Id ⟶ ℤ.
∀v1,v2:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List).
∀p:ℤ × Cmd + pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List) + pv11_p1_Ballot_Num().
((e1 <loc e2)
⇒ p ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e1)
⇒ v1 ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e1)
⇒ v2 ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e2)
⇒ case p
of inl(x) =>
let s,c = x
in let bnum1,active1,proposals1 = v1 in
let bnum2,active2,proposals2 = v2 in
(¬↑(pv11_p1_in_domain(Cmd) s proposals1)) ⇒ (↑(pv11_p1_in_domain(Cmd) s proposals2))
| inr(x) =>
True)
BY
{ MemoryMemory }
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. 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. x2 : ℤ@i
29. x3 : Cmd@i
30. s1 : pv11_p1_Ballot_Num()@i
31. s3 : 𝔹@i
32. s4 : (ℤ × Cmd) List@i
33. ¬(x2 ∈ map(λx.(fst(x));s4))
34. e : E@i
35. e1 ≤loc e @i
36. (e <loc e2)@i
37. inl <x2, x3> ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e)@i
38. <s1, s3, s4> ∈
Memory-loc-class(pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + ...;λloc.{pv11_p1_init_leader(Cmd)
loc};... (+) ...)(e)@i
39. ¬(x2 ∈ map(λx.(fst(x));s4))@i
⊢ ∃y@0:ℤ × Cmd. ((y@0 ∈ pv11_p1_add_if_new() pv11_p1_same_proposal(Cmd) <x2, x3> s4) ∧ (x2 = (fst(y@0)) ∈ ℤ))
2
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 : pv11_p1_Ballot_Num()@i
23. x4 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
24. (e1 <loc e2)@i
25. inr (inl <x3, x4>) ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(\000Ce1)@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. x1 : ℤ@i
29. x2 : Cmd@i
30. s1 : pv11_p1_Ballot_Num()@i
31. s3 : 𝔹@i
32. s4 : (ℤ × Cmd) List@i
33. ¬(x1 ∈ map(λx.(fst(x));s4))
34. e : E@i
35. e1 ≤loc e @i
36. (e <loc e2)@i
37. inl <x1, x2> ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e)@i
38. <s1, s3, s4> ∈
Memory-loc-class(pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + ...;λloc.{pv11_p1_init_leader(Cmd)
loc};... (+) ...)(e)@i
39. ¬(x1 ∈ map(λx.(fst(x));s4))@i
⊢ ∃y@0:ℤ × Cmd. ((y@0 ∈ pv11_p1_add_if_new() pv11_p1_same_proposal(Cmd) <x1, x2> s4) ∧ (x1 = (fst(y@0)) ∈ ℤ))
3
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. y1 : pv11_p1_Ballot_Num()@i
23. (e1 <loc e2)@i
24. inr inr y1 ∈ 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. x1 : ℤ@i
28. x2 : Cmd@i
29. s1 : pv11_p1_Ballot_Num()@i
30. s3 : 𝔹@i
31. s4 : (ℤ × Cmd) List@i
32. ¬(x1 ∈ map(λx.(fst(x));s4))
33. e : E@i
34. e1 ≤loc e @i
35. (e <loc e2)@i
36. inl <x1, x2> ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e)@i
37. <s1, s3, s4> ∈
Memory-loc-class(pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + ...;λloc.{pv11_p1_init_leader(Cmd)
loc};... (+) ...)(e)@i
38. ¬(x1 ∈ map(λx.(fst(x));s4))@i
⊢ ∃y@0:ℤ × Cmd. ((y@0 ∈ pv11_p1_add_if_new() pv11_p1_same_proposal(Cmd) <x1, x2> s4) ∧ (x1 = (fst(y@0)) ∈ ℤ))
4
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. x7 : ℤ@i
23. x8 : Cmd@i
24. (e1 <loc e2)@i
25. inl <x7, x8> ∈ 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. x5 : ℤ@i
29. x6 : Cmd@i
30. x3 : ℤ@i
31. x4 : 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. ¬(x3 ∈ map(λx.(fst(x));s8))
39. e : E@i
40. e' : E@i
41. e1 ≤loc e @i
42. (e <loc e')@i
43. (e' <loc e2)@i
44. inl <x5, x6> ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e)@i
45. <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
46. inl <x3, x4> ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e')@i
47. <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
48. ¬(x5 ∈ map(λx.(fst(x));s10))
49. y1 : ℤ
50. y2 : Cmd
51. (<y1, y2> ∈ s8)
52. x5 = y1 ∈ ℤ
⊢ ∃y@0:ℤ × Cmd. ((y@0 ∈ pv11_p1_add_if_new() pv11_p1_same_proposal(Cmd) <x3, x4> s8) ∧ (x5 = (fst(y@0)) ∈ ℤ))
5
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. x6 : pv11_p1_Ballot_Num()@i
23. x7 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
24. (e1 <loc e2)@i
25. inr (inl <x6, x7>) ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(\000Ce1)@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. x4 : ℤ@i
29. x5 : Cmd@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. ¬(x2 ∈ map(λx.(fst(x));s8))
39. e : E@i
40. e' : E@i
41. e1 ≤loc e @i
42. (e <loc e')@i
43. (e' <loc e2)@i
44. inl <x4, x5> ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e)@i
45. <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
46. inl <x2, x3> ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e')@i
47. <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
48. ¬(x4 ∈ map(λx.(fst(x));s10))
49. y1 : ℤ
50. y2 : Cmd
51. (<y1, y2> ∈ s8)
52. x4 = y1 ∈ ℤ
⊢ ∃y@0:ℤ × Cmd. ((y@0 ∈ pv11_p1_add_if_new() pv11_p1_same_proposal(Cmd) <x2, x3> s8) ∧ (x4 = (fst(y@0)) ∈ ℤ))
6
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. y1 : pv11_p1_Ballot_Num()@i
23. (e1 <loc e2)@i
24. inr inr y1 ∈ 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. x4 : ℤ@i
28. x5 : Cmd@i
29. x2 : ℤ@i
30. x3 : Cmd@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. ¬(x2 ∈ map(λx.(fst(x));s8))
38. e : E@i
39. e' : E@i
40. e1 ≤loc e @i
41. (e <loc e')@i
42. (e' <loc e2)@i
43. inl <x4, x5> ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e)@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. ¬(x4 ∈ map(λx.(fst(x));s10))
48. y2 : ℤ
49. y3 : Cmd
50. (<y2, y3> ∈ s8)
51. x4 = y2 ∈ ℤ
⊢ ∃y@0:ℤ × Cmd. ((y@0 ∈ pv11_p1_add_if_new() pv11_p1_same_proposal(Cmd) <x2, x3> s8) ∧ (x4 = (fst(y@0)) ∈ ℤ))
7
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. x5 : ℤ@i
23. x6 : Cmd@i
24. (e1 <loc e2)@i
25. inl <x5, x6> ∈ 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. x3 : ℤ@i
29. x4 : Cmd@i
30. x7 : pv11_p1_Ballot_Num()@i
31. x8 : (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. inl <x3, x4> ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e)@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 <x7, x8>) ∈ 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. x7 = s3 ∈ pv11_p1_Ballot_Num()
48. ¬(x3 ∈ map(λx.(fst(x));s10))
49. y1 : ℤ
50. y2 : Cmd
51. (<y1, y2> ∈ s8)
52. x3 = y1 ∈ ℤ
⊢ ∃y:ℤ × Cmd. ((y ∈ pv11_p1_update_proposals(Cmd) s8 (pv11_p1_pmax(Cmd;ldrs_uid) x8)) ∧ (x3 = (fst(y)) ∈ ℤ))
8
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. x6 : pv11_p1_Ballot_Num()@i
23. x7 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
24. (e1 <loc e2)@i
25. inr (inl <x6, x7>) ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(\000Ce1)@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. x2 : ℤ@i
29. x3 : Cmd@i
30. x4 : pv11_p1_Ballot_Num()@i
31. x5 : (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. inl <x2, x3> ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e)@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 <x4, x5>) ∈ 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. x4 = s3 ∈ pv11_p1_Ballot_Num()
48. ¬(x2 ∈ map(λx.(fst(x));s10))
49. y1 : ℤ
50. y2 : Cmd
51. (<y1, y2> ∈ s8)
52. x2 = y1 ∈ ℤ
⊢ ∃y:ℤ × Cmd. ((y ∈ pv11_p1_update_proposals(Cmd) s8 (pv11_p1_pmax(Cmd;ldrs_uid) x5)) ∧ (x2 = (fst(y)) ∈ ℤ))
9
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))
Latex:
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)). \mforall{}e1,e2:E.
\mforall{}ldrs$_{uid}$:Id {}\mrightarrow{} \mBbbZ{}. \mforall{}v1,v2:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List).
\mforall{}p:\mBbbZ{} \mtimes{} Cmd + pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List) + pv11\_p1\_Ballot\_Num().
((e1 <loc e2)
{}\mRightarrow{} p \mmember{}
pv11\_p1\_propose'base(Cmd;f) (+) pv11\_p1\_adopted'base(Cmd;f) (+) pv11\_p1\_preempted'base(Cmd;f)(
e1)
{}\mRightarrow{} v1 \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e1)
{}\mRightarrow{} v2 \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e2)
{}\mRightarrow{} case p
of inl(x) =>
let s,c = x
in let bnum1,active1,proposals1 = v1 in
let bnum2,active2,proposals2 = v2 in
(\mneg{}\muparrow{}(pv11\_p1\_in\_domain(Cmd) s proposals1)) {}\mRightarrow{} (\muparrow{}(pv11\_p1\_in\_domain(Cmd) s proposals2))
| inr(x) =>
True)
By
Latex:
MemoryMemory
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