Step
*
1
1
of Lemma
pv11_p1_A2
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. ldrs_uid : Id ─→ ℤ@i
15. bnum1 : pv11_p1_Ballot_Num()@i
16. bnum2 : pv11_p1_Ballot_Num()@i
17. accepted1 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
18. b : pv11_p1_Ballot_Num()@i
19. pv : ℤ × Cmd@i
20. loc : Id@i
21. Inj(Id;ℤ;ldrs_uid)@i
22. ¬↑first(e)@i
23. <bnum1, accepted1> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(pred(e))@i
24. ¬(<b, pv> ∈ accepted1)@i
25. ↑(pv11_p1_leq_bnum(ldrs_uid) bnum1 bnum2)
26. l : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
27. (<b, pv> ∈ l)
28. <bnum2, accepted1 @ l> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e)
⊢ b = bnum2 ∈ pv11_p1_Ballot_Num()
BY
{ ... }
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. e : E@i
14. ¬↑first(e)
15. ldrs_uid : Id ─→ ℤ@i
16. bnum1 : pv11_p1_Ballot_Num()@i
17. bnum2 : pv11_p1_Ballot_Num()@i
18. accepted1 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
19. b : pv11_p1_Ballot_Num()@i
20. pv : ℤ × Cmd@i
21. loc : Id@i
22. Inj(Id;ℤ;ldrs_uid)@i
23. ¬False@i
24. <bnum1, accepted1>
= pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))
∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
25. ¬(<b, pv> ∈ accepted1)@i
26. ↑(pv11_p1_leq_bnum(ldrs_uid) bnum1 bnum2)
27. l : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
28. (<b, pv> ∈ l)
29. ↑e ∈b pv11_p1_p1a'base(Cmd;f)
⊢ (<bnum2, accepted1 @ l>
= ((pv11_p1_on_p1a(Cmd;ldrs_uid) o pv11_p1_p1a'base(Cmd;f))
|| (pv11_p1_on_p2a(Cmd;ldrs_uid) o pv11_p1_p2a'base(Cmd;f))@e
pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))
∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
⇒ (b = bnum2 ∈ pv11_p1_Ballot_Num())
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. 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. pv : ℤ × Cmd@i
22. loc : Id@i
23. Inj(Id;ℤ;ldrs_uid)@i
24. ¬False@i
25. <bnum1, accepted1>
= pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))
∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
26. ¬(<b, pv> ∈ accepted1)@i
27. ↑(pv11_p1_leq_bnum(ldrs_uid) bnum1 bnum2)
28. l : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
29. (<b, pv> ∈ l)
30. ↑e ∈b pv11_p1_p2a'base(Cmd;f)
⊢ (<bnum2, accepted1 @ l>
= ((pv11_p1_on_p1a(Cmd;ldrs_uid) o pv11_p1_p1a'base(Cmd;f))
|| (pv11_p1_on_p2a(Cmd;ldrs_uid) o pv11_p1_p2a'base(Cmd;f))@e
pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))
∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
⇒ (b = bnum2 ∈ pv11_p1_Ballot_Num())
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. e : E@i
14. ¬↑first(e)
15. ¬↑e ∈b pv11_p1_p2a'base(Cmd;f)
16. ¬↑e ∈b pv11_p1_p1a'base(Cmd;f)
17. ldrs_uid : Id ─→ ℤ@i
18. bnum1 : pv11_p1_Ballot_Num()@i
19. bnum2 : pv11_p1_Ballot_Num()@i
20. accepted1 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
21. b : pv11_p1_Ballot_Num()@i
22. pv : ℤ × Cmd@i
23. loc : Id@i
24. Inj(Id;ℤ;ldrs_uid)@i
25. ¬False@i
26. <bnum1, accepted1>
= pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))
∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
27. ¬(<b, pv> ∈ accepted1)@i
28. ↑(pv11_p1_leq_bnum(ldrs_uid) bnum1 bnum2)
29. l : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
30. (<b, pv> ∈ l)
⊢ (<bnum2, accepted1 @ l>
= pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))
∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
⇒ (b = bnum2 ∈ pv11_p1_Ballot_Num())
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. e : E@i
14. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i
15. bnum1 : pv11\_p1\_Ballot\_Num()@i
16. bnum2 : pv11\_p1\_Ballot\_Num()@i
17. accepted1 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
18. b : pv11\_p1\_Ballot\_Num()@i
19. pv : \mBbbZ{} \mtimes{} Cmd@i
20. loc : Id@i
21. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i
22. \mneg{}\muparrow{}first(e)@i
23. <bnum1, accepted1> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(pred(e))@i
24. \mneg{}(<b, pv> \mmember{} accepted1)@i
25. \muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) bnum1 bnum2)
26. l : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List
27. (<b, pv> \mmember{} l)
28. <bnum2, accepted1 @ l> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e)
\mvdash{} b = bnum2
By
Latex:
(All(\mbackslash{}i.(RWO "pv11\_p1\_AcceptorState-classrel" i THENA Auto))\mcdot{}
THEN (RWO "pv11\_p1\_acc\_state\_fun\_eq" (-1) THENA Auto)
THEN (RWO "member-disjoint-union-comb-bool" (-1) THENA Auto)
THEN MoveToConcl (-1)
THEN Repeat (AutoSplit)
THEN Try ((RWO "classfun-res-disjoint-union-comb-as-parallel-eclass1" 0
THENA (Auto THEN ProveFunctional THEN Auto)
)))
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