Step
*
1
of Lemma
pv11_p1_ldr_mem_adopted
.....antecedent.....
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. v1 : pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)@i
17. v2 : pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)@i
18. p : ℤ × Cmd + pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List) + pv11_p1_Ballot_Num()@i
19. (e1 <loc e2)@i
20. p ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e1)@i
21. v1 ∈
Memory-loc-class(pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + ...;λloc.{pv11_p1_init_leader(Cmd)
loc};... (+) ...)(e1)
22. v2 ∈
Memory-loc-class(pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + ...;λloc.{pv11_p1_init_leader(Cmd)
loc};... (+) ...)(e2)
⊢ ∀a:ℤ × Cmd + pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List) + pv11_p1_Ballot_Num().
∀s:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List). ∀e:E.
(e1 ≤loc e
⇒ (e <loc e2)
⇒ a ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e)
⇒ s ∈
Memory-loc-class(pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + ...;λloc.{...
loc};... (+) ...)(e)
⇒ (λ2v2.case a
of inl(x) =>
True
| inr(x) =>
case x
of inl(x) =>
let bnum,pvals = x
in let bnum1,active1,proposals1 = s in
let bnum2,active2,proposals2 = v2 in
(bnum1 = bnum ∈ pv11_p1_Ballot_Num())
⇒ (∀s:ℤ. ∀b:pv11_p1_Ballot_Num(). ∀c:Cmd.
(((<s, c> ∈ proposals1) ∨ (<b, s, c> ∈ pvals)) ⇒ (↑(pv11_p1_in_domain(Cmd) s proposals2))))
| inr(x) =>
True
(pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + pv11_p1_when_preempted(Cmd;ldrs_uid) loc(e) a
s)))
BY
{ GenMemoryTrans2 }
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. x2 : ℤ@i
23. x3 : Cmd@i
24. (e1 <loc e2)@i
25. inl <x2, x3> ∈ 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. x4 : pv11_p1_Ballot_Num()@i
29. x5 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
30. s1 : pv11_p1_Ballot_Num()@i
31. s3 : 𝔹@i
32. s4 : (ℤ × Cmd) List@i
33. e : E@i
34. e1 ≤loc e @i
35. (e <loc e2)@i
36. inr (inl <x4, x5>) ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(\000Ce)@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. x4 = s1 ∈ pv11_p1_Ballot_Num()
39. s1 = x4 ∈ pv11_p1_Ballot_Num()@i
40. s : ℤ@i
41. b : pv11_p1_Ballot_Num()@i
42. c : Cmd@i
43. (<s, c> ∈ s4)@i
⊢ ∃y:ℤ × Cmd. ((y ∈ pv11_p1_update_proposals(Cmd) s4 (pv11_p1_pmax(Cmd;ldrs_uid) x5)) ∧ (s = (fst(y)) ∈ 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. x2 : ℤ@i
23. x3 : Cmd@i
24. (e1 <loc e2)@i
25. inl <x2, x3> ∈ 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. x4 : pv11_p1_Ballot_Num()@i
29. x5 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
30. s1 : pv11_p1_Ballot_Num()@i
31. s3 : 𝔹@i
32. s4 : (ℤ × Cmd) List@i
33. e : E@i
34. e1 ≤loc e @i
35. (e <loc e2)@i
36. inr (inl <x4, x5>) ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(\000Ce)@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. x4 = s1 ∈ pv11_p1_Ballot_Num()
39. s1 = x4 ∈ pv11_p1_Ballot_Num()@i
40. s : ℤ@i
41. b : pv11_p1_Ballot_Num()@i
42. c : Cmd@i
43. (<b, s, c> ∈ x5)@i
⊢ ∃y:ℤ × Cmd. ((y ∈ pv11_p1_update_proposals(Cmd) s4 (pv11_p1_pmax(Cmd;ldrs_uid) x5)) ∧ (s = (fst(y)) ∈ 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. x4 : pv11_p1_Ballot_Num()@i
23. x5 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
24. (e1 <loc e2)@i
25. inr (inl <x4, x5>) ∈ 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 : pv11_p1_Ballot_Num()@i
29. x3 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
30. s1 : pv11_p1_Ballot_Num()@i
31. s3 : 𝔹@i
32. s4 : (ℤ × Cmd) List@i
33. e : E@i
34. e1 ≤loc e @i
35. (e <loc e2)@i
36. inr (inl <x2, x3>) ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(\000Ce)@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. x2 = s1 ∈ pv11_p1_Ballot_Num()
39. s1 = x2 ∈ pv11_p1_Ballot_Num()@i
40. s : ℤ@i
41. b : pv11_p1_Ballot_Num()@i
42. c : Cmd@i
43. (<s, c> ∈ s4)@i
⊢ ∃y:ℤ × Cmd. ((y ∈ pv11_p1_update_proposals(Cmd) s4 (pv11_p1_pmax(Cmd;ldrs_uid) x3)) ∧ (s = (fst(y)) ∈ 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. x4 : pv11_p1_Ballot_Num()@i
23. x5 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
24. (e1 <loc e2)@i
25. inr (inl <x4, x5>) ∈ 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 : pv11_p1_Ballot_Num()@i
29. x3 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
30. s1 : pv11_p1_Ballot_Num()@i
31. s3 : 𝔹@i
32. s4 : (ℤ × Cmd) List@i
33. e : E@i
34. e1 ≤loc e @i
35. (e <loc e2)@i
36. inr (inl <x2, x3>) ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(\000Ce)@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. x2 = s1 ∈ pv11_p1_Ballot_Num()
39. s1 = x2 ∈ pv11_p1_Ballot_Num()@i
40. s : ℤ@i
41. b : pv11_p1_Ballot_Num()@i
42. c : Cmd@i
43. (<b, s, c> ∈ x3)@i
⊢ ∃y:ℤ × Cmd. ((y ∈ pv11_p1_update_proposals(Cmd) s4 (pv11_p1_pmax(Cmd;ldrs_uid) x3)) ∧ (s = (fst(y)) ∈ 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. 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. x1 : pv11_p1_Ballot_Num()@i
28. x2 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
29. s1 : pv11_p1_Ballot_Num()@i
30. s3 : 𝔹@i
31. s4 : (ℤ × Cmd) List@i
32. e : E@i
33. e1 ≤loc e @i
34. (e <loc e2)@i
35. inr (inl <x1, x2>) ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(\000Ce)@i
36. <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
37. x1 = s1 ∈ pv11_p1_Ballot_Num()
38. s1 = x1 ∈ pv11_p1_Ballot_Num()@i
39. s : ℤ@i
40. b : pv11_p1_Ballot_Num()@i
41. c : Cmd@i
42. (<s, c> ∈ s4)@i
⊢ ∃y:ℤ × Cmd. ((y ∈ pv11_p1_update_proposals(Cmd) s4 (pv11_p1_pmax(Cmd;ldrs_uid) x2)) ∧ (s = (fst(y)) ∈ 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. 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. x1 : pv11_p1_Ballot_Num()@i
28. x2 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
29. s1 : pv11_p1_Ballot_Num()@i
30. s3 : 𝔹@i
31. s4 : (ℤ × Cmd) List@i
32. e : E@i
33. e1 ≤loc e @i
34. (e <loc e2)@i
35. inr (inl <x1, x2>) ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(\000Ce)@i
36. <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
37. x1 = s1 ∈ pv11_p1_Ballot_Num()
38. s1 = x1 ∈ pv11_p1_Ballot_Num()@i
39. s : ℤ@i
40. b : pv11_p1_Ballot_Num()@i
41. c : Cmd@i
42. (<b, s, c> ∈ x2)@i
⊢ ∃y:ℤ × Cmd. ((y ∈ pv11_p1_update_proposals(Cmd) s4 (pv11_p1_pmax(Cmd;ldrs_uid) x2)) ∧ (s = (fst(y)) ∈ ℤ))
Latex:
Latex:
.....antecedent.....
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. v1 : pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List)@i
17. v2 : pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List)@i
18. p : \mBbbZ{} \mtimes{} Cmd + pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List) + pv11\_p1\_Ballot\_Num()@i
19. (e1 <loc e2)@i
20. p \mmember{}
pv11\_p1\_propose'base(Cmd;f) (+) pv11\_p1\_adopted'base(Cmd;f) (+) pv11\_p1\_preempted'base(Cmd;f)(
e1)@i
21. v1 \mmember{}
Memory-loc-class(pv11\_p1\_on\_propose(Cmd) + pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid}\000C$) + ...;...;...)(e1)
22. v2 \mmember{}
Memory-loc-class(pv11\_p1\_on\_propose(Cmd) + pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid}\000C$) + ...;...;...)(e2)
\mvdash{} \mforall{}a:\mBbbZ{} \mtimes{} Cmd + pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List) + pv11\_p1\_Ballot\_Num().
\mforall{}s:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List). \mforall{}e:E.
(e1 \mleq{}loc e
{}\mRightarrow{} (e <loc e2)
{}\mRightarrow{} a \mmember{}
pv11\_p1\_propose'base(Cmd;f) (+) pv11\_p1\_adopted'base(Cmd;f) (+) ...(e)
{}\mRightarrow{} s \mmember{}
Memory-loc-class(pv11\_p1\_on\_propose(Cmd) + pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid\mbackslash{}f\000Cf7d$) + ...;...;...)(
e)
{}\mRightarrow{} (\mlambda{}\msubtwo{}v2.case a
of inl(x) =>
True
| inr(x) =>
case x
of inl(x) =>
let bnum,pvals = x
in let bnum1,active1,proposals1 = s in
let bnum2,active2,proposals2 = v2 in
(bnum1 = bnum)
{}\mRightarrow{} (\mforall{}s:\mBbbZ{}. \mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}c:Cmd.
(((<s, c> \mmember{} proposals1) \mvee{} (<b, s, c> \mmember{} pvals))
{}\mRightarrow{} (\muparrow{}(pv11\_p1\_in\_domain(Cmd) s proposals2))))
| inr(x) =>
True
(pv11\_p1\_on\_propose(Cmd) + pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid}$) + ...
loc(e)
a
s)))
By
Latex:
GenMemoryTrans2
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