Step
*
1
1
of Lemma
pv11_p1_append_news_iff2
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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs_uid : Id ─→ ℤ@i
16. reps : bag(Id)@i
17. pv : pv11_p1_Ballot_Num() × ℤ × Cmd@i
18. Inj(Id;ℤ;ldrs_uid)@i
19. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
20. u : pv11_p1_Ballot_Num() × ℤ × Cmd@i
21. v : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
22. ∀L1:(pv11_p1_Ballot_Num() × ℤ × Cmd) List
((∀x∈L1 @ v.↓∃e:E. (x ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e))))
⇒ ((pv ∈ accumulate (with value a and list item x):
pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) x a
over list:
v
with starting value:
L1))
⇐⇒ (pv ∈ L1) ∨ (pv ∈ v)))@i
23. L1 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
24. (∀x∈L1 @ [u / v].↓∃e:E. (x ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e))))@i
25. (pv ∈ accumulate (with value a and list item x):
pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) x a
over list:
v
with starting value:
pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) u L1))@i
⊢ (pv ∈ L1) ∨ (pv ∈ [u / v])
BY
{ (RWO "-4" (-1) THENA Auto) }
1
.....rewrite subgoal.....
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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs_uid : Id ─→ ℤ@i
16. reps : bag(Id)@i
17. pv : pv11_p1_Ballot_Num() × ℤ × Cmd@i
18. Inj(Id;ℤ;ldrs_uid)@i
19. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
20. u : pv11_p1_Ballot_Num() × ℤ × Cmd@i
21. v : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
22. ∀L1:(pv11_p1_Ballot_Num() × ℤ × Cmd) List
((∀x∈L1 @ v.↓∃e:E. (x ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e))))
⇒ ((pv ∈ accumulate (with value a and list item x):
pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) x a
over list:
v
with starting value:
L1))
⇐⇒ (pv ∈ L1) ∨ (pv ∈ v)))@i
23. L1 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
24. (∀x∈L1 @ [u / v].↓∃e:E. (x ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e))))@i
25. (pv ∈ accumulate (with value a and list item x):
pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) x a
over list:
v
with starting value:
pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) u L1))@i
⊢ (∀x∈(pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) u L1) @ v.
↓∃e:E. (x ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e))))
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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs_uid : Id ─→ ℤ@i
16. reps : bag(Id)@i
17. pv : pv11_p1_Ballot_Num() × ℤ × Cmd@i
18. Inj(Id;ℤ;ldrs_uid)@i
19. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
20. u : pv11_p1_Ballot_Num() × ℤ × Cmd@i
21. v : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
22. ∀L1:(pv11_p1_Ballot_Num() × ℤ × Cmd) List
((∀x∈L1 @ v.↓∃e:E. (x ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e))))
⇒ ((pv ∈ accumulate (with value a and list item x):
pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) x a
over list:
v
with starting value:
L1))
⇐⇒ (pv ∈ L1) ∨ (pv ∈ v)))@i
23. L1 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
24. (∀x∈L1 @ [u / v].↓∃e:E. (x ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e))))@i
25. (pv ∈ pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) u L1) ∨ (pv ∈ v)
⊢ (pv ∈ L1) ∨ (pv ∈ [u / v])
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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i
16. reps : bag(Id)@i
17. pv : pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd@i
18. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i
19. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i
20. u : pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd@i
21. v : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
22. \mforall{}L1:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List
((\mforall{}x\mmember{}L1 @ v.\mdownarrow{}\mexists{}e:E. (x \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es\000C;e))))
{}\mRightarrow{} ((pv \mmember{} accumulate (with value a and list item x):
pv11\_p1\_add\_if\_new() pv11\_p1\_same\_pvalue(Cmd) x a
over list:
v
with starting value:
L1))
\mLeftarrow{}{}\mRightarrow{} (pv \mmember{} L1) \mvee{} (pv \mmember{} v)))@i
23. L1 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
24. (\mforall{}x\mmember{}L1 @ [u / v].\mdownarrow{}\mexists{}e:E. (x \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f\000C;es;e))))@i
25. (pv \mmember{} accumulate (with value a and list item x):
pv11\_p1\_add\_if\_new() pv11\_p1\_same\_pvalue(Cmd) x a
over list:
v
with starting value:
pv11\_p1\_add\_if\_new() pv11\_p1\_same\_pvalue(Cmd) u L1))@i
\mvdash{} (pv \mmember{} L1) \mvee{} (pv \mmember{} [u / v])
By
Latex:
(RWO "-4" (-1) THENA Auto)
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