Step
*
2
of Lemma
pv11_p1_ldr_mem_preempted
.....antecedent.....
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. v1 : pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)@i
17. v2 : pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)@i
18. p : ℤ × Cmd + pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List) + pv11_p1_Ballot_Num()@i
19. (e1 <loc e2)@i
20. p ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e1)@i
21. v1 ∈
Memory-loc-class(pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + ...;λloc.{pv11_p1_init_leader(Cmd)
loc};... (+) ...)(e1)
22. v2 ∈
Memory-loc-class(pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + ...;λloc.{pv11_p1_init_leader(Cmd)
loc};... (+) ...)(e2)
⊢ ∀a1,a2:ℤ × Cmd + pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List) + pv11_p1_Ballot_Num().
∀s1,s2:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List). ∀e,e':E.
(e1 ≤loc e
⇒ (e <loc e')
⇒ (e' <loc e2)
⇒ a1 ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e)
⇒ s1 ∈
Memory-loc-class(pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + ...;λloc.{...
loc};... (+) ...)(e)
⇒ a2 ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e')
⇒ s2 ∈
Memory-loc-class(pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + ...;λloc.{...
loc};... (+) ...)(e')
⇒ (λ2v2.let bnum1,active1,proposals1 = s1 in
let bnum2,active2,proposals2 = v2 in
case a1
of inl(x) =>
True
| inr(x) =>
case x
of inl(x) =>
True
| inr(bnum) =>
(↑(bnum1 < bnum)) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) (pv11_p1_upd_bnum() bnum loc(e1)) bnum2))
s2)
⇒ (λ2v2.let bnum1,active1,proposals1 = s1 in
let bnum2,active2,proposals2 = v2 in
case a1
of inl(x) =>
True
| inr(x) =>
case x
of inl(x) =>
True
| inr(bnum) =>
(↑(bnum1 < bnum)) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) (pv11_p1_upd_bnum() bnum loc(e1)) bnum2))
(pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + pv11_p1_when_preempted(Cmd;ldrs_uid) loc(e2) a2
s2)))
BY
{ ReduceAutoMemory }
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. v1 : pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)@i
17. v2 : pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)@i
18. p : ℤ × Cmd + pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List) + pv11_p1_Ballot_Num()@i
19. (e1 <loc e2)@i
20. p ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e1)@i
21. v1 ∈
Memory-loc-class(pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + ...;λloc.{pv11_p1_init_leader(Cmd)
loc};... (+) ...)(e1)
22. v2 ∈
Memory-loc-class(pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + ...;λloc.{pv11_p1_init_leader(Cmd)
loc};... (+) ...)(e2)
23. a1 : ℤ × Cmd + pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List) + pv11_p1_Ballot_Num()@i
24. a2 : ℤ × Cmd + pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List) + pv11_p1_Ballot_Num()@i
25. s1 : pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)@i
26. s2 : pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)@i
27. e : E@i
28. e' : E@i
29. e1 ≤loc e @i
30. (e <loc e')@i
31. (e' <loc e2)@i
32. a1 ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e)@i
33. s1 ∈
Memory-loc-class(pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + ...;λloc.{pv11_p1_init_leader(Cmd)
loc};... (+) ...)(e)@i
34. a2 ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(e')@i
35. s2 ∈
Memory-loc-class(pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + ...;λloc.{pv11_p1_init_leader(Cmd)
loc};... (+) ...)(e')@i
36. let bnum1,active1,proposals1 = s1 in
let bnum2,active2,proposals2 = s2 in
case a1
of inl(x) =>
True
| inr(x) =>
case x
of inl(x) =>
True
| inr(bnum) =>
(↑(bnum1 < bnum)) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) (pv11_p1_upd_bnum() bnum loc(e1)) bnum2))@18
⊢ let bnum1,active1,proposals1 = s1 in
let bnum2,active2,proposals2 = pv11_p1_on_propose(Cmd) + pv11_p1_when_adopted(Cmd;ldrs_uid) + ...
loc(e2)
a2
s2 in
case a1
of inl(x) =>
True
| inr(x) =>
case x
of inl(x) =>
True
| inr(bnum) =>
(↑(bnum1 < bnum)) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) (pv11_p1_upd_bnum() bnum loc(e1)) bnum2))
Latex:
Latex:
.....antecedent.....
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. v1 : pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List)@i
17. v2 : pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List)@i
18. p : \mBbbZ{} \mtimes{} Cmd + pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List) + pv11\_p1\_Ballot\_Num()@i
19. (e1 <loc e2)@i
20. p \mmember{}
pv11\_p1\_propose'base(Cmd;f) (+) pv11\_p1\_adopted'base(Cmd;f) (+) pv11\_p1\_preempted'base(Cmd;f)(
e1)@i
21. v1 \mmember{}
Memory-loc-class(pv11\_p1\_on\_propose(Cmd) + pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid}\000C$) + ...;...;...)(e1)
22. v2 \mmember{}
Memory-loc-class(pv11\_p1\_on\_propose(Cmd) + pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid}\000C$) + ...;...;...)(e2)
\mvdash{} \mforall{}a1,a2:\mBbbZ{} \mtimes{} Cmd + pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List) + pv11\_p1\_Ballot\_Num().
\mforall{}s1,s2:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List). \mforall{}e,e':E.
(e1 \mleq{}loc e
{}\mRightarrow{} (e <loc e')
{}\mRightarrow{} (e' <loc e2)
{}\mRightarrow{} a1 \mmember{}
pv11\_p1\_propose'base(Cmd;f) (+) pv11\_p1\_adopted'base(Cmd;f) (+) ...(e)
{}\mRightarrow{} s1 \mmember{}
Memory-loc-class(pv11\_p1\_on\_propose(Cmd) + pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid\mbackslash{}f\000Cf7d$) + ...;...;...)(
e)
{}\mRightarrow{} a2 \mmember{}
pv11\_p1\_propose'base(Cmd;f) (+) pv11\_p1\_adopted'base(Cmd;f) (+) ...(e')
{}\mRightarrow{} s2 \mmember{}
Memory-loc-class(pv11\_p1\_on\_propose(Cmd) + pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid\mbackslash{}f\000Cf7d$) + ...;...;...)(
e')
{}\mRightarrow{} (\mlambda{}\msubtwo{}v2.let bnum1,active1,proposals1 = s1 in
let bnum2,active2,proposals2 = v2 in
case a1
of inl(x) =>
True
| inr(x) =>
case x
of inl(x) =>
True
| inr(bnum) =>
(\muparrow{}(bnum1 < bnum))
{}\mRightarrow{} (\muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) (pv11\_p1\_upd\_bnum() bnum loc(e1))\000C bnum2))
s2)
{}\mRightarrow{} (\mlambda{}\msubtwo{}v2.let bnum1,active1,proposals1 = s1 in
let bnum2,active2,proposals2 = v2 in
case a1
of inl(x) =>
True
| inr(x) =>
case x
of inl(x) =>
True
| inr(bnum) =>
(\muparrow{}(bnum1 < bnum))
{}\mRightarrow{} (\muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) (pv11\_p1\_upd\_bnum() bnum loc(e1))\000C bnum2))
(pv11\_p1\_on\_propose(Cmd) + pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid}$) + ...
loc(e2)
a2
s2)))
By
Latex:
ReduceAutoMemory
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