Step
*
2
of Lemma
pv11_p1_ldr_fun_mem_adopted
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. pvals : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
17. s : ℤ@i
18. c : Cmd@i
19. (e1 <loc e2)@i
20. <fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)), pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e1)@i
21. (↓(<s, c> ∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)))))
∨ (∃b:pv11_p1_Ballot_Num(). (↓(<b, s, c> ∈ pvals)))@i
22. case inr (inl <fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)), pvals>)
of inl(x) =>
True
| inr(x) =>
case x
of inl(x) =>
let bnum,pvals = x
in let bnum1,active1,proposals1 = pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1) in
let bnum2,active2,proposals2 = pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2) 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_in_domain(Cmd) s (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2)))))
BY
{ (AllReduce THEN RepeatFor 4 (UsePairEta [1] (-1)) THEN Auto) }
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. pvals : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
17. s : ℤ@i
18. c : Cmd@i
19. (e1 <loc e2)@i
20. <fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)), pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e1)@i
21. (↓(<s, c> ∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)))))
∨ (∃b:pv11_p1_Ballot_Num(). (↓(<b, s, c> ∈ pvals)))@i
22. ((fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)))
= (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)))
∈ pv11_p1_Ballot_Num())
⇒ (∀s:ℤ. ∀b:pv11_p1_Ballot_Num(). ∀c:Cmd.
(((<s, c> ∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)))) ∨ (<b, s, c> ∈ pvals))
⇒ (↑(pv11_p1_in_domain(Cmd) s (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2))))))))
⊢ ↑(pv11_p1_in_domain(Cmd) s (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2)))))
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. pvals : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
17. s : \mBbbZ{}@i
18. c : Cmd@i
19. (e1 <loc e2)@i
20. <fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1)), pvals> \mmember{} pv11\_p1\_ado\000Cpted'base(Cmd;f)(e1)@i
21. (\mdownarrow{}(<s, c> \mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1)))))
\mvee{} (\mexists{}b:pv11\_p1\_Ballot\_Num(). (\mdownarrow{}(<b, s, c> \mmember{} pvals)))@i
22. case inr (inl <fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1)), pvals>\000C)
of inl(x) =>
True
| inr(x) =>
case x
of inl(x) =>
let bnum,pvals = x
in let bnum1,active1,proposals1 = pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es\000C;e1) in
let bnum2,active2,proposals2 = pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es\000C;e2) 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
\mvdash{} \muparrow{}(pv11\_p1\_in\_domain(Cmd) s (snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;\000Ces;e2)))))
By
Latex:
(AllReduce THEN RepeatFor 4 (UsePairEta [1] (-1)) THEN Auto)
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