Step
*
1
1
2
1
1
of Lemma
pv11_p1_A2
.....truecase.....
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. e : E@i
14. ¬↑first(e)
15. ¬↑e ∈b pv11_p1_p1a'base(Cmd;f)
16. ldrs_uid : Id ─→ ℤ@i
17. bnum1 : pv11_p1_Ballot_Num()@i
18. bnum2 : pv11_p1_Ballot_Num()@i
19. accepted1 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
20. b : pv11_p1_Ballot_Num()@i
21. p1 : ℤ@i
22. p2 : Cmd@i
23. loc : Id@i
24. Inj(Id;ℤ;ldrs_uid)@i
25. ¬False@i
26. bnum1 = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))) ∈ pv11_p1_Ballot_Num()
27. accepted1 = (snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))) ∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
28. ¬(<b, p1, p2> ∈ accepted1)@i
29. ↑(pv11_p1_leq_bnum(ldrs_uid) bnum1 bnum2)
30. l : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
31. (<b, p1, p2> ∈ l)
32. ↑e ∈b Base(``pv11_p1 p2a``)
33. pv11_p1_p2a'base(Cmd;f)@e ~ msgval(e)
34. header(e) = ``pv11_p1 p2a`` ∈ Name
35. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
36. v1 : Id@i
37. v3 : pv11_p1_Ballot_Num()@i
38. v5 : ℤ@i
39. v6 : Cmd@i
40. (fst(msgval(e))) = v1 ∈ Id
41. (snd(msgval(e))) = <v3, v5, v6> ∈ (pv11_p1_Ballot_Num() × ℤ × Cmd)
42. bnum2 = (pv11_p1_max_bnum(ldrs_uid) v3 bnum1) ∈ pv11_p1_Ballot_Num()
43. (accepted1 @ l) = (pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) <v3, v5, v6> accepted1) ∈ ((pv11_p1_Ballot_Num() × \000Cℤ × Cmd) List)
44. ↑(pv11_p1_leq_bnum(ldrs_uid) bnum1 v3)
⊢ b = bnum2 ∈ pv11_p1_Ballot_Num()
BY
{ (RepUR ``pv11_p1_add_if_new`` (-2) THEN (SplitOnHypITE (-2) THENA Auto)) }
1
.....truecase.....
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. e : E@i
14. ¬↑first(e)
15. ¬↑e ∈b pv11_p1_p1a'base(Cmd;f)
16. ldrs_uid : Id ─→ ℤ@i
17. bnum1 : pv11_p1_Ballot_Num()@i
18. bnum2 : pv11_p1_Ballot_Num()@i
19. accepted1 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
20. b : pv11_p1_Ballot_Num()@i
21. p1 : ℤ@i
22. p2 : Cmd@i
23. loc : Id@i
24. Inj(Id;ℤ;ldrs_uid)@i
25. ¬False@i
26. bnum1 = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))) ∈ pv11_p1_Ballot_Num()
27. accepted1 = (snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))) ∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
28. ¬(<b, p1, p2> ∈ accepted1)@i
29. ↑(pv11_p1_leq_bnum(ldrs_uid) bnum1 bnum2)
30. l : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
31. (<b, p1, p2> ∈ l)
32. ↑e ∈b Base(``pv11_p1 p2a``)
33. pv11_p1_p2a'base(Cmd;f)@e ~ msgval(e)
34. header(e) = ``pv11_p1 p2a`` ∈ Name
35. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
36. v1 : Id@i
37. v3 : pv11_p1_Ballot_Num()@i
38. v5 : ℤ@i
39. v6 : Cmd@i
40. (fst(msgval(e))) = v1 ∈ Id
41. (snd(msgval(e))) = <v3, v5, v6> ∈ (pv11_p1_Ballot_Num() × ℤ × Cmd)
42. bnum2 = (pv11_p1_max_bnum(ldrs_uid) v3 bnum1) ∈ pv11_p1_Ballot_Num()
43. (accepted1 @ l) = accepted1 ∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
44. ↑(pv11_p1_leq_bnum(ldrs_uid) bnum1 v3)
45. (∃zzc∈accepted1. ↑(pv11_p1_same_pvalue(Cmd) <v3, v5, v6> zzc))
⊢ b = bnum2 ∈ pv11_p1_Ballot_Num()
2
.....falsecase.....
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. e : E@i
14. ¬↑first(e)
15. ¬↑e ∈b pv11_p1_p1a'base(Cmd;f)
16. ldrs_uid : Id ─→ ℤ@i
17. bnum1 : pv11_p1_Ballot_Num()@i
18. bnum2 : pv11_p1_Ballot_Num()@i
19. accepted1 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
20. b : pv11_p1_Ballot_Num()@i
21. p1 : ℤ@i
22. p2 : Cmd@i
23. loc : Id@i
24. Inj(Id;ℤ;ldrs_uid)@i
25. ¬False@i
26. bnum1 = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))) ∈ pv11_p1_Ballot_Num()
27. accepted1 = (snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))) ∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
28. ¬(<b, p1, p2> ∈ accepted1)@i
29. ↑(pv11_p1_leq_bnum(ldrs_uid) bnum1 bnum2)
30. l : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
31. (<b, p1, p2> ∈ l)
32. ↑e ∈b Base(``pv11_p1 p2a``)
33. pv11_p1_p2a'base(Cmd;f)@e ~ msgval(e)
34. header(e) = ``pv11_p1 p2a`` ∈ Name
35. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
36. v1 : Id@i
37. v3 : pv11_p1_Ballot_Num()@i
38. v5 : ℤ@i
39. v6 : Cmd@i
40. (fst(msgval(e))) = v1 ∈ Id
41. (snd(msgval(e))) = <v3, v5, v6> ∈ (pv11_p1_Ballot_Num() × ℤ × Cmd)
42. bnum2 = (pv11_p1_max_bnum(ldrs_uid) v3 bnum1) ∈ pv11_p1_Ballot_Num()
43. (accepted1 @ l) = (accepted1 @ [<v3, v5, v6>]) ∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
44. ↑(pv11_p1_leq_bnum(ldrs_uid) bnum1 v3)
45. ¬(∃zzc∈accepted1. ↑(pv11_p1_same_pvalue(Cmd) <v3, v5, v6> zzc))
⊢ b = bnum2 ∈ pv11_p1_Ballot_Num()
Latex:
Latex:
.....truecase.....
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. e : E@i
14. \mneg{}\muparrow{}first(e)
15. \mneg{}\muparrow{}e \mmember{}\msubb{} pv11\_p1\_p1a'base(Cmd;f)
16. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i
17. bnum1 : pv11\_p1\_Ballot\_Num()@i
18. bnum2 : pv11\_p1\_Ballot\_Num()@i
19. accepted1 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
20. b : pv11\_p1\_Ballot\_Num()@i
21. p1 : \mBbbZ{}@i
22. p2 : Cmd@i
23. loc : Id@i
24. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i
25. \mneg{}False@i
26. bnum1 = (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;pred(e))))
27. accepted1 = (snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;pred(e))))
28. \mneg{}(<b, p1, p2> \mmember{} accepted1)@i
29. \muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) bnum1 bnum2)
30. l : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List
31. (<b, p1, p2> \mmember{} l)
32. \muparrow{}e \mmember{}\msubb{} Base(``pv11\_p1 p2a``)
33. pv11\_p1\_p2a'base(Cmd;f)@e \msim{} msgval(e)
34. header(e) = ``pv11\_p1 p2a``
35. has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd)
36. v1 : Id@i
37. v3 : pv11\_p1\_Ballot\_Num()@i
38. v5 : \mBbbZ{}@i
39. v6 : Cmd@i
40. (fst(msgval(e))) = v1
41. (snd(msgval(e))) = <v3, v5, v6>
42. bnum2 = (pv11\_p1\_max\_bnum(ldrs$_{uid}$) v3 bnum1)
43. (accepted1 @ l) = (pv11\_p1\_add\_if\_new() pv11\_p1\_same\_pvalue(Cmd) <v3, v5, v6> accepted1)
44. \muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) bnum1 v3)
\mvdash{} b = bnum2
By
Latex:
(RepUR ``pv11\_p1\_add\_if\_new`` (-2) THEN (SplitOnHypITE (-2) THENA Auto))
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