Step
*
1
11
of Lemma
pv11_p1_ldr_state_eq2
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_adopted'base(Cmd;f)
15. ¬↑pred(e) ∈b pv11_p1_propose'base(Cmd;f)
16. ¬↑first(e)
17. ldrs_uid : Id ⟶ ℤ@i
18. v : pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)@i
19. if pred(e) ∈b pv11_p1_preempted'base(Cmd;f)
then ∃x:pv11_p1_Ballot_Num()
∃s:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)
(x ∈ pv11_p1_preempted'base(Cmd;f)(pred(e))
∧ s ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
∧ (v = (pv11_p1_when_preempted(Cmd;ldrs_uid) loc(e) x s) ∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List))))
else v ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
fi
supposing v
= if pred(e) ∈b pv11_p1_preempted'base(Cmd;f)
then pv11_p1_when_preempted(Cmd;ldrs_uid) loc(e) pv11_p1_preempted'base(Cmd;f)@pred(e)
pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;pred(e))
else pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;pred(e))
fi
∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List))
20. v
= if pred(e) ∈b pv11_p1_preempted'base(Cmd;f)
then pv11_p1_when_preempted(Cmd;ldrs_uid) loc(e) pv11_p1_preempted'base(Cmd;f)@pred(e)
pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;pred(e))
else pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;pred(e))
fi
∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List))
supposing if pred(e) ∈b pv11_p1_preempted'base(Cmd;f)
then ∃x:pv11_p1_Ballot_Num()
∃s:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)
(x ∈ pv11_p1_preempted'base(Cmd;f)(pred(e))
∧ s ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
∧ (v = (pv11_p1_when_preempted(Cmd;ldrs_uid) loc(e) x s) ∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List))))
else v ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(pred(e))
fi
21. ff ∈ 𝔹
22. ff ∈ 𝔹
23. ff ∈ 𝔹
24. pred(e) ∈b pv11_p1_preempted'base(Cmd;f) ∈ 𝔹
25. ↑pred(e) ∈b pv11_p1_preempted'base(Cmd;f)
26. e1 : E@i
27. v1 : pv11_p1_Ballot_Num()@i
28. v2 : pv11_p1_Ballot_Num()@i
29. v1 ∈ pv11_p1_preempted'base(Cmd;f)(e1)@i
30. v2 ∈ pv11_p1_preempted'base(Cmd;f)(e1)@i
⊢ v1 = v2 ∈ pv11_p1_Ballot_Num()
BY
{ (SingleVal `e1' THEN Auto) }
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\_adopted'base(Cmd;f)
15. \mneg{}\muparrow{}pred(e) \mmember{}\msubb{} pv11\_p1\_propose'base(Cmd;f)
16. \mneg{}\muparrow{}first(e)
17. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i
18. v : pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List)@i
19. if pred(e) \mmember{}\msubb{} pv11\_p1\_preempted'base(Cmd;f)
then \mexists{}x:pv11\_p1\_Ballot\_Num()
\mexists{}s:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List)
(x \mmember{} pv11\_p1\_preempted'base(Cmd;f)(pred(e))
\mwedge{} s \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(pred(e))
\mwedge{} (v = (pv11\_p1\_when\_preempted(Cmd;ldrs$_{uid}$) loc(e) x s)))
else v \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(pred(e))
fi
supposing v
= if pred(e) \mmember{}\msubb{} pv11\_p1\_preempted'base(Cmd;f)
then pv11\_p1\_when\_preempted(Cmd;ldrs$_{uid}$) loc(e) pv11\_p1\_preempted'bas\000Ce(Cmd;f)@pred(e)
pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;pred(e))
else pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;pred(e))
fi
20. v
= if pred(e) \mmember{}\msubb{} pv11\_p1\_preempted'base(Cmd;f)
then pv11\_p1\_when\_preempted(Cmd;ldrs$_{uid}$) loc(e) pv11\_p1\_preempted'bas\000Ce(Cmd;f)@pred(e)
pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;pred(e))
else pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;pred(e))
fi
supposing if pred(e) \mmember{}\msubb{} pv11\_p1\_preempted'base(Cmd;f)
then \mexists{}x:pv11\_p1\_Ballot\_Num()
\mexists{}s:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List)
(x \mmember{} pv11\_p1\_preempted'base(Cmd;f)(pred(e))
\mwedge{} s \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(pred(e))
\mwedge{} (v = (pv11\_p1\_when\_preempted(Cmd;ldrs$_{uid}$) loc(e) x s)))
else v \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(pred(e))
fi
21. ff \mmember{} \mBbbB{}
22. ff \mmember{} \mBbbB{}
23. ff \mmember{} \mBbbB{}
24. pred(e) \mmember{}\msubb{} pv11\_p1\_preempted'base(Cmd;f) \mmember{} \mBbbB{}
25. \muparrow{}pred(e) \mmember{}\msubb{} pv11\_p1\_preempted'base(Cmd;f)
26. e1 : E@i
27. v1 : pv11\_p1\_Ballot\_Num()@i
28. v2 : pv11\_p1\_Ballot\_Num()@i
29. v1 \mmember{} pv11\_p1\_preempted'base(Cmd;f)(e1)@i
30. v2 \mmember{} pv11\_p1\_preempted'base(Cmd;f)(e1)@i
\mvdash{} v1 = v2
By
Latex:
(SingleVal `e1' THEN Auto)
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