Step
*
2
of Lemma
pv11_p1_ldr_state_eq
1. Cmd : {T:Type| valueall-type(T)}
2. f : pv11_p1_headers_type{i:l}(Cmd)
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))
13. e : E
14. ¬↑pred(e) ∈b pv11_p1_propose'base(Cmd;f)
15. ¬↑first(e)
16. ldrs_uid : Id ─→ ℤ
17. v : pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)
18. ↑pred(e) ∈b pv11_p1_adopted'base(Cmd;f)
19. b : pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)
20. b ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
21. (∃a1:ℤ × Cmd
(a1 ∈ pv11_p1_propose'base(Cmd;f)(pred(e))
∧ (v = (pv11_p1_on_propose(Cmd) loc(e) a1 b) ∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)))))
∨ (∃a2:pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
(a2 ∈ pv11_p1_adopted'base(Cmd;f)(pred(e))
∧ (v = (pv11_p1_when_adopted(Cmd;ldrs_uid) loc(e) a2 b) ∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)))))
∨ (∃a3:pv11_p1_Ballot_Num()
(a3 ∈ pv11_p1_preempted'base(Cmd;f)(pred(e))
∧ (v = (pv11_p1_when_preempted(Cmd;ldrs_uid) loc(e) a3 b) ∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)))))
⊢ ↓∃x:pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
∃s:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)
(x ∈ pv11_p1_adopted'base(Cmd;f)(pred(e))
∧ s ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
∧ (v = (pv11_p1_when_adopted(Cmd;ldrs_uid) loc(e) x s) ∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List))))
BY
{ DProdsAndUnions }
1
1. Cmd : {T:Type| valueall-type(T)}
2. f : pv11_p1_headers_type{i:l}(Cmd)
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))
13. e : E
14. ¬↑pred(e) ∈b pv11_p1_propose'base(Cmd;f)
15. ¬↑first(e)
16. ldrs_uid : Id ─→ ℤ
17. v1 : pv11_p1_Ballot_Num()
18. v3 : 𝔹
19. v4 : (ℤ × Cmd) List
20. ↑pred(e) ∈b pv11_p1_adopted'base(Cmd;f)
21. b1 : pv11_p1_Ballot_Num()
22. b3 : 𝔹
23. b4 : (ℤ × Cmd) List
24. <b1, b3, b4> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
25. a2 : ℤ
26. a3 : Cmd
27. <a2, a3> ∈ pv11_p1_propose'base(Cmd;f)(pred(e))
28. <v1, v3, v4> = (pv11_p1_on_propose(Cmd) loc(e) <a2, a3> <b1, b3, b4>) ∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)\000C)
⊢ ↓∃x:pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
∃s:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)
(x ∈ pv11_p1_adopted'base(Cmd;f)(pred(e))
∧ s ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
∧ (<v1, v3, v4> = (pv11_p1_when_adopted(Cmd;ldrs_uid) loc(e) x s) ∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List))))
2
1. Cmd : {T:Type| valueall-type(T)}
2. f : pv11_p1_headers_type{i:l}(Cmd)
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))
13. e : E
14. ¬↑pred(e) ∈b pv11_p1_propose'base(Cmd;f)
15. ¬↑first(e)
16. ldrs_uid : Id ─→ ℤ
17. v1 : pv11_p1_Ballot_Num()
18. v3 : 𝔹
19. v4 : (ℤ × Cmd) List
20. ↑pred(e) ∈b pv11_p1_adopted'base(Cmd;f)
21. b1 : pv11_p1_Ballot_Num()
22. b3 : 𝔹
23. b4 : (ℤ × Cmd) List
24. <b1, b3, b4> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
25. a3 : pv11_p1_Ballot_Num()
26. a4 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
27. <a3, a4> ∈ pv11_p1_adopted'base(Cmd;f)(pred(e))
28. <v1, v3, v4> = (pv11_p1_when_adopted(Cmd;ldrs_uid) loc(e) <a3, a4> <b1, b3, b4>) ∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ ×\000C Cmd) List))
⊢ ↓∃x:pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
∃s:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)
(x ∈ pv11_p1_adopted'base(Cmd;f)(pred(e))
∧ s ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
∧ (<v1, v3, v4> = (pv11_p1_when_adopted(Cmd;ldrs_uid) loc(e) x s) ∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List))))
3
1. Cmd : {T:Type| valueall-type(T)}
2. f : pv11_p1_headers_type{i:l}(Cmd)
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))
13. e : E
14. ¬↑pred(e) ∈b pv11_p1_propose'base(Cmd;f)
15. ¬↑first(e)
16. ldrs_uid : Id ─→ ℤ
17. v1 : pv11_p1_Ballot_Num()
18. v3 : 𝔹
19. v4 : (ℤ × Cmd) List
20. ↑pred(e) ∈b pv11_p1_adopted'base(Cmd;f)
21. b1 : pv11_p1_Ballot_Num()
22. b3 : 𝔹
23. b4 : (ℤ × Cmd) List
24. <b1, b3, b4> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
25. a3 : pv11_p1_Ballot_Num()
26. a3 ∈ pv11_p1_preempted'base(Cmd;f)(pred(e))
27. <v1, v3, v4> = (pv11_p1_when_preempted(Cmd;ldrs_uid) loc(e) a3 <b1, b3, b4>) ∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd\000C) List))
⊢ ↓∃x:pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
∃s:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)
(x ∈ pv11_p1_adopted'base(Cmd;f)(pred(e))
∧ s ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
∧ (<v1, v3, v4> = (pv11_p1_when_adopted(Cmd;ldrs_uid) loc(e) x s) ∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List))))
Latex:
Latex:
1. Cmd : \{T:Type| valueall-type(T)\}
2. f : pv11\_p1\_headers\_type\{i:l\}(Cmd)
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))
13. e : E
14. \mneg{}\muparrow{}pred(e) \mmember{}\msubb{} pv11\_p1\_propose'base(Cmd;f)
15. \mneg{}\muparrow{}first(e)
16. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}
17. v : pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List)
18. \muparrow{}pred(e) \mmember{}\msubb{} pv11\_p1\_adopted'base(Cmd;f)
19. b : pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List)
20. b \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(pred(e))
21. (\mexists{}a1:\mBbbZ{} \mtimes{} Cmd
(a1 \mmember{} pv11\_p1\_propose'base(Cmd;f)(pred(e)) \mwedge{} (v = (pv11\_p1\_on\_propose(Cmd) loc(e) a1 b))))
\mvee{} (\mexists{}a2:pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)
(a2 \mmember{} pv11\_p1\_adopted'base(Cmd;f)(pred(e))
\mwedge{} (v = (pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid}$) loc(e) a2 b))))
\mvee{} (\mexists{}a3:pv11\_p1\_Ballot\_Num()
(a3 \mmember{} pv11\_p1\_preempted'base(Cmd;f)(pred(e))
\mwedge{} (v = (pv11\_p1\_when\_preempted(Cmd;ldrs$_{uid}$) loc(e) a3 b))))
\mvdash{} \mdownarrow{}\mexists{}x:pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)
\mexists{}s:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List)
(x \mmember{} pv11\_p1\_adopted'base(Cmd;f)(pred(e))
\mwedge{} s \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(pred(e))
\mwedge{} (v = (pv11\_p1\_when\_adopted(Cmd;ldrs$_{uid}$) loc(e) x s)))
By
Latex:
DProdsAndUnions
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