Step
*
1
2
1
of Lemma
pv11_p1_ldr_proposal3
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. ldrs_uid : Id ─→ ℤ@i
14. e : E@i
15. ∀e':E
((e' < e)
⇒ (∀v1:pv11_p1_Ballot_Num(). ∀v3:𝔹. ∀v4:(ℤ × Cmd) List. ∀s:ℤ. ∀c:Cmd.
(<v1, v3, v4> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e')
⇒ (<s, c> ∈ v4)
⇒ (∃e1:E
∃bnum:pv11_p1_Ballot_Num()
∃active:𝔹
∃proposals:(ℤ × Cmd) List
((e1 <loc e')
∧ <bnum, active, proposals> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e1)
∧ (∀e'':E. ∀x:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List).
((e1 <loc e'')
⇒ e'' ≤loc e'
⇒ x ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e'')
⇒ (<s, c> ∈ snd(snd(x)))))
∧ ((<s, c> ∈ pv11_p1_propose'base(Cmd;f)(e1) ∧ (¬↑(pv11_p1_in_domain(Cmd) s proposals)))
∨ (∃pvals:(pv11_p1_Ballot_Num() × ℤ × Cmd) List
∃b:pv11_p1_Ballot_Num()
(<bnum, pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e1)
∧ (<b, s, c> ∈ pvals)
∧ (∀b':pv11_p1_Ballot_Num(). ∀c':Cmd.
((<b', s, c'> ∈ pvals) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) b' b))))))))))))
16. v1 : pv11_p1_Ballot_Num()@i
17. v3 : 𝔹@i
18. v4 : (ℤ × Cmd) List@i
19. s : ℤ@i
20. c : Cmd@i
21. <v1, v3, v4> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e)@i
22. (<s, c> ∈ v4)@i
23. x : ℤ × Cmd
24. s1 : pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)
25. x ∈ pv11_p1_propose'base(Cmd;f)(pred(e))
26. s1 ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
27. <v1, v3, v4> = (pv11_p1_on_propose(Cmd) loc(e) x s1) ∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List))
28. ¬↑first(e)
29. ↑pred(e) ∈b pv11_p1_propose'base(Cmd;f)
⊢ ∃e':E
∃bnum:pv11_p1_Ballot_Num()
∃active:𝔹
∃proposals:(ℤ × Cmd) List
((e' <loc e)
∧ <bnum, active, proposals> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e')
∧ (∀e'':E. ∀x:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List).
((e' <loc e'') ⇒ e'' ≤loc e ⇒ x ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e'') ⇒ (<s, c> ∈ snd(snd(x)))))
∧ ((<s, c> ∈ pv11_p1_propose'base(Cmd;f)(e') ∧ (¬↑(pv11_p1_in_domain(Cmd) s proposals)))
∨ (∃pvals:(pv11_p1_Ballot_Num() × ℤ × Cmd) List
∃b:pv11_p1_Ballot_Num()
(<bnum, pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e')
∧ (<b, s, c> ∈ pvals)
∧ (∀b':pv11_p1_Ballot_Num(). ∀c':Cmd. ((<b', s, c'> ∈ pvals) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) b' b))))))))
BY
{ (DProdsAndUnions
THEN RepUR ``pv11_p1_on_propose let`` (-3)
THEN (SplitOnHypITE (-3) THENA Auto)
THEN Repeat (SimpEqPairs)) }
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. ldrs_uid : Id ─→ ℤ@i
14. e : E@i
15. ∀e':E
((e' < e)
⇒ (∀v1:pv11_p1_Ballot_Num(). ∀v3:𝔹. ∀v4:(ℤ × Cmd) List. ∀s:ℤ. ∀c:Cmd.
(<v1, v3, v4> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e')
⇒ (<s, c> ∈ v4)
⇒ (∃e1:E
∃bnum:pv11_p1_Ballot_Num()
∃active:𝔹
∃proposals:(ℤ × Cmd) List
((e1 <loc e')
∧ <bnum, active, proposals> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e1)
∧ (∀e'':E. ∀x:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List).
((e1 <loc e'')
⇒ e'' ≤loc e'
⇒ x ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e'')
⇒ (<s, c> ∈ snd(snd(x)))))
∧ ((<s, c> ∈ pv11_p1_propose'base(Cmd;f)(e1) ∧ (¬↑(pv11_p1_in_domain(Cmd) s proposals)))
∨ (∃pvals:(pv11_p1_Ballot_Num() × ℤ × Cmd) List
∃b:pv11_p1_Ballot_Num()
(<bnum, pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e1)
∧ (<b, s, c> ∈ pvals)
∧ (∀b':pv11_p1_Ballot_Num(). ∀c':Cmd.
((<b', s, c'> ∈ pvals) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) b' b))))))))))))
16. v1 : pv11_p1_Ballot_Num()@i
17. v3 : 𝔹@i
18. v4 : (ℤ × Cmd) List@i
19. s : ℤ@i
20. c : Cmd@i
21. <v1, v3, v4> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e)@i
22. (<s, c> ∈ v4)@i
23. x1 : ℤ
24. x2 : Cmd
25. s2 : pv11_p1_Ballot_Num()
26. s4 : 𝔹
27. s5 : (ℤ × Cmd) List
28. <x1, x2> ∈ pv11_p1_propose'base(Cmd;f)(pred(e))
29. <s2, s4, s5> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
30. v1 = s2 ∈ pv11_p1_Ballot_Num()
31. v3 = s4
32. v4 = s5 ∈ ((ℤ × Cmd) List)
33. ¬↑first(e)
34. ↑pred(e) ∈b pv11_p1_propose'base(Cmd;f)
35. ↑(pv11_p1_in_domain(Cmd) x1 s5)
⊢ ∃e':E
∃bnum:pv11_p1_Ballot_Num()
∃active:𝔹
∃proposals:(ℤ × Cmd) List
((e' <loc e)
∧ <bnum, active, proposals> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e')
∧ (∀e'':E. ∀x:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List).
((e' <loc e'') ⇒ e'' ≤loc e ⇒ x ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e'') ⇒ (<s, c> ∈ snd(snd(x)))))
∧ ((<s, c> ∈ pv11_p1_propose'base(Cmd;f)(e') ∧ (¬↑(pv11_p1_in_domain(Cmd) s proposals)))
∨ (∃pvals:(pv11_p1_Ballot_Num() × ℤ × Cmd) List
∃b:pv11_p1_Ballot_Num()
(<bnum, pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e')
∧ (<b, s, c> ∈ pvals)
∧ (∀b':pv11_p1_Ballot_Num(). ∀c':Cmd. ((<b', s, c'> ∈ pvals) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) b' b))))))))
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. ldrs_uid : Id ─→ ℤ@i
14. e : E@i
15. ∀e':E
((e' < e)
⇒ (∀v1:pv11_p1_Ballot_Num(). ∀v3:𝔹. ∀v4:(ℤ × Cmd) List. ∀s:ℤ. ∀c:Cmd.
(<v1, v3, v4> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e')
⇒ (<s, c> ∈ v4)
⇒ (∃e1:E
∃bnum:pv11_p1_Ballot_Num()
∃active:𝔹
∃proposals:(ℤ × Cmd) List
((e1 <loc e')
∧ <bnum, active, proposals> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e1)
∧ (∀e'':E. ∀x:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List).
((e1 <loc e'')
⇒ e'' ≤loc e'
⇒ x ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e'')
⇒ (<s, c> ∈ snd(snd(x)))))
∧ ((<s, c> ∈ pv11_p1_propose'base(Cmd;f)(e1) ∧ (¬↑(pv11_p1_in_domain(Cmd) s proposals)))
∨ (∃pvals:(pv11_p1_Ballot_Num() × ℤ × Cmd) List
∃b:pv11_p1_Ballot_Num()
(<bnum, pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e1)
∧ (<b, s, c> ∈ pvals)
∧ (∀b':pv11_p1_Ballot_Num(). ∀c':Cmd.
((<b', s, c'> ∈ pvals) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) b' b))))))))))))
16. v1 : pv11_p1_Ballot_Num()@i
17. v3 : 𝔹@i
18. v4 : (ℤ × Cmd) List@i
19. s : ℤ@i
20. c : Cmd@i
21. <v1, v3, v4> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e)@i
22. (<s, c> ∈ v4)@i
23. x1 : ℤ
24. x2 : Cmd
25. s2 : pv11_p1_Ballot_Num()
26. s4 : 𝔹
27. s5 : (ℤ × Cmd) List
28. <x1, x2> ∈ pv11_p1_propose'base(Cmd;f)(pred(e))
29. <s2, s4, s5> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
30. v1 = s2 ∈ pv11_p1_Ballot_Num()
31. v3 = s4
32. v4 = (pv11_p1_add_if_new() pv11_p1_same_proposal(Cmd) <x1, x2> s5) ∈ ((ℤ × Cmd) List)
33. ¬↑first(e)
34. ↑pred(e) ∈b pv11_p1_propose'base(Cmd;f)
35. ¬↑(pv11_p1_in_domain(Cmd) x1 s5)
⊢ ∃e':E
∃bnum:pv11_p1_Ballot_Num()
∃active:𝔹
∃proposals:(ℤ × Cmd) List
((e' <loc e)
∧ <bnum, active, proposals> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e')
∧ (∀e'':E. ∀x:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List).
((e' <loc e'') ⇒ e'' ≤loc e ⇒ x ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e'') ⇒ (<s, c> ∈ snd(snd(x)))))
∧ ((<s, c> ∈ pv11_p1_propose'base(Cmd;f)(e') ∧ (¬↑(pv11_p1_in_domain(Cmd) s proposals)))
∨ (∃pvals:(pv11_p1_Ballot_Num() × ℤ × Cmd) List
∃b:pv11_p1_Ballot_Num()
(<bnum, pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e')
∧ (<b, s, c> ∈ pvals)
∧ (∀b':pv11_p1_Ballot_Num(). ∀c':Cmd. ((<b', s, c'> ∈ pvals) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) b' b))))))))
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. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i
14. e : E@i
15. \mforall{}e':E
((e' < e)
{}\mRightarrow{} (\mforall{}v1:pv11\_p1\_Ballot\_Num(). \mforall{}v3:\mBbbB{}. \mforall{}v4:(\mBbbZ{} \mtimes{} Cmd) List. \mforall{}s:\mBbbZ{}. \mforall{}c:Cmd.
(<v1, v3, v4> \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e')
{}\mRightarrow{} (<s, c> \mmember{} v4)
{}\mRightarrow{} (\mexists{}e1:E
\mexists{}bnum:pv11\_p1\_Ballot\_Num()
\mexists{}active:\mBbbB{}
\mexists{}proposals:(\mBbbZ{} \mtimes{} Cmd) List
((e1 <loc e')
\mwedge{} <bnum, active, proposals> \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid\mbackslash{}ff7\000Cd$;f)(e1)
\mwedge{} (\mforall{}e'':E. \mforall{}x:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List).
((e1 <loc e'')
{}\mRightarrow{} e'' \mleq{}loc e'
{}\mRightarrow{} x \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e'')
{}\mRightarrow{} (<s, c> \mmember{} snd(snd(x)))))
\mwedge{} ((<s, c> \mmember{} pv11\_p1\_propose'base(Cmd;f)(e1)
\mwedge{} (\mneg{}\muparrow{}(pv11\_p1\_in\_domain(Cmd) s proposals)))
\mvee{} (\mexists{}pvals:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List
\mexists{}b:pv11\_p1\_Ballot\_Num()
(<bnum, pvals> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e1)
\mwedge{} (<b, s, c> \mmember{} pvals)
\mwedge{} (\mforall{}b':pv11\_p1\_Ballot\_Num(). \mforall{}c':Cmd.
((<b', s, c'> \mmember{} pvals)
{}\mRightarrow{} (\muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) b' b)))))))\000C)))))
16. v1 : pv11\_p1\_Ballot\_Num()@i
17. v3 : \mBbbB{}@i
18. v4 : (\mBbbZ{} \mtimes{} Cmd) List@i
19. s : \mBbbZ{}@i
20. c : Cmd@i
21. <v1, v3, v4> \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e)@i
22. (<s, c> \mmember{} v4)@i
23. x : \mBbbZ{} \mtimes{} Cmd
24. s1 : pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List)
25. x \mmember{} pv11\_p1\_propose'base(Cmd;f)(pred(e))
26. s1 \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(pred(e))
27. <v1, v3, v4> = (pv11\_p1\_on\_propose(Cmd) loc(e) x s1)
28. \mneg{}\muparrow{}first(e)
29. \muparrow{}pred(e) \mmember{}\msubb{} pv11\_p1\_propose'base(Cmd;f)
\mvdash{} \mexists{}e':E
\mexists{}bnum:pv11\_p1\_Ballot\_Num()
\mexists{}active:\mBbbB{}
\mexists{}proposals:(\mBbbZ{} \mtimes{} Cmd) List
((e' <loc e)
\mwedge{} <bnum, active, proposals> \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e')
\mwedge{} (\mforall{}e'':E. \mforall{}x:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List).
((e' <loc e'')
{}\mRightarrow{} e'' \mleq{}loc e
{}\mRightarrow{} x \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e'')
{}\mRightarrow{} (<s, c> \mmember{} snd(snd(x)))))
\mwedge{} ((<s, c> \mmember{} pv11\_p1\_propose'base(Cmd;f)(e') \mwedge{} (\mneg{}\muparrow{}(pv11\_p1\_in\_domain(Cmd) s proposals)))
\mvee{} (\mexists{}pvals:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List
\mexists{}b:pv11\_p1\_Ballot\_Num()
(<bnum, pvals> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e')
\mwedge{} (<b, s, c> \mmember{} pvals)
\mwedge{} (\mforall{}b':pv11\_p1\_Ballot\_Num(). \mforall{}c':Cmd.
((<b', s, c'> \mmember{} pvals) {}\mRightarrow{} (\muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) b'\000C b))))))))
By
Latex:
(DProdsAndUnions
THEN RepUR ``pv11\_p1\_on\_propose let`` (-3)
THEN (SplitOnHypITE (-3) THENA Auto)
THEN Repeat (SimpEqPairs))
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