Step
*
1
1
of Lemma
pv11_p1_adopted_prior
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. ¬↑pred(e) ∈b pv11_p1_propose'base(Cmd;f)
15. ¬↑first(e)
16. ldrs_uid : Id ⟶ ℤ@i
17. bnum : pv11_p1_Ballot_Num()@i
18. active : 𝔹@i
19. proposals : (ℤ × Cmd) List@i
20. pvals : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
21. b' : pv11_p1_Ballot_Num()@i
22. s : ℤ@i
23. c : Cmd@i
24. c' : Cmd@i
25. ¬False@i
26. (<s, c> ∈ proposals)@i
27. <bnum, pvals> ∈ pv11_p1_adopted'base(Cmd;f)(pred(e))@i
28. (<b', s, c'> ∈ pvals)@i
29. ↑pred(e) ∈b pv11_p1_adopted'base(Cmd;f)
30. x : pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)@i
31. s1 : pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)@i
32. x ∈ pv11_p1_adopted'base(Cmd;f)(pred(e))@i
33. s1 ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))@i
34. <bnum, active, proposals>
= (pv11_p1_when_adopted(Cmd;ldrs_uid) loc(e) x s1)
∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List))@i
⊢ ↓∃b:pv11_p1_Ballot_Num(). ((<b, s, c> ∈ pvals) ∧ (↑(pv11_p1_leq_bnum(ldrs_uid) b' b)))
BY
{ (SingleValTerm ⌜pred(e)⌝⋅
THEN (HypSubst (-1) (-2) THENA Auto)
THEN ThinVar `x'
THEN DProdsAndUnions
THEN RepUR ``pv11_p1_when_adopted let`` (-1)
THEN (SplitOnHypITE (-1) THENA Auto)
THEN RepeatFor 2 (SimpEqPairs)
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. e : E@i
14. ¬↑pred(e) ∈b pv11_p1_propose'base(Cmd;f)
15. ¬↑first(e)
16. ldrs_uid : Id ⟶ ℤ@i
17. bnum : pv11_p1_Ballot_Num()@i
18. active : 𝔹@i
19. proposals : (ℤ × Cmd) List@i
20. pvals : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
21. b' : pv11_p1_Ballot_Num()@i
22. s : ℤ@i
23. c : Cmd@i
24. c' : Cmd@i
25. ¬False@i
26. (<s, c> ∈ proposals)@i
27. <bnum, pvals> ∈ pv11_p1_adopted'base(Cmd;f)(pred(e))@i
28. (<b', s, c'> ∈ pvals)@i
29. ↑pred(e) ∈b pv11_p1_adopted'base(Cmd;f)
30. s2 : pv11_p1_Ballot_Num()@i
31. s4 : 𝔹@i
32. s5 : (ℤ × Cmd) List@i
33. <s2, s4, s5> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))@i
34. bnum = s2 ∈ pv11_p1_Ballot_Num()
35. active = tt
36. proposals = (pv11_p1_update_proposals(Cmd) s5 (pv11_p1_pmax(Cmd;ldrs_uid) pvals)) ∈ ((ℤ × Cmd) List)
37. bnum = s2 ∈ pv11_p1_Ballot_Num()
⊢ ↓∃b:pv11_p1_Ballot_Num(). ((<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. e : E@i
14. \mneg{}\muparrow{}pred(e) \mmember{}\msubb{} pv11\_p1\_propose'base(Cmd;f)
15. \mneg{}\muparrow{}first(e)
16. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i
17. bnum : pv11\_p1\_Ballot\_Num()@i
18. active : \mBbbB{}@i
19. proposals : (\mBbbZ{} \mtimes{} Cmd) List@i
20. pvals : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
21. b' : pv11\_p1\_Ballot\_Num()@i
22. s : \mBbbZ{}@i
23. c : Cmd@i
24. c' : Cmd@i
25. \mneg{}False@i
26. (<s, c> \mmember{} proposals)@i
27. <bnum, pvals> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(pred(e))@i
28. (<b', s, c'> \mmember{} pvals)@i
29. \muparrow{}pred(e) \mmember{}\msubb{} pv11\_p1\_adopted'base(Cmd;f)
30. x : pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)@i
31. s1 : pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List)@i
32. x \mmember{} pv11\_p1\_adopted'base(Cmd;f)(pred(e))@i
33. s1 \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(pred(e))@i
34. <bnum, active, proposals> = (pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid}$) loc(e) x\000C s1)@i
\mvdash{} \mdownarrow{}\mexists{}b:pv11\_p1\_Ballot\_Num(). ((<b, s, c> \mmember{} pvals) \mwedge{} (\muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}\mbackslash{}f\000Cf24) b' b)))
By
Latex:
(SingleValTerm \mkleeneopen{}pred(e)\mkleeneclose{}\mcdot{}
THEN (HypSubst (-1) (-2) THENA Auto)
THEN ThinVar `x'
THEN DProdsAndUnions
THEN RepUR ``pv11\_p1\_when\_adopted let`` (-1)
THEN (SplitOnHypITE (-1) THENA Auto)
THEN RepeatFor 2 (SimpEqPairs)
THEN Auto)
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