Step
*
1
of Lemma
pv11_p1_ldr_state_propose_pred
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. bnum : pv11_p1_Ballot_Num()@i
17. active : 𝔹@i
18. proposals : (ℤ × Cmd) List@i
19. s : ℤ@i
20. c : Cmd@i
21. (e1 <loc e2)@i
22. <bnum, active, proposals> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e1)@i
23. header(e1) = [propose] ∈ Name@i
24. has-es-info-type(es;e1;f;ℤ × Cmd)@i
25. <s, c> = msgval(e1) ∈ (ℤ × Cmd)@i
26. ¬↑(pv11_p1_in_domain(Cmd) s proposals)@i
⊢ ∃e:{e:E| ¬↑first(e)}
((e1 = pred(e) ∈ E) ∧ e ≤loc e2 ∧ <bnum, active, pv11_p1_add_if_new() pv11_p1_same_proposal(Cmd) <s, c> proposals> ∈\000C pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e))
BY
{ ((InstLemma `es-pred-exists-between2` [⌜parm{i'}⌝;⌜es⌝;⌜e1⌝;⌜e2⌝]⋅ THENA Auto)
THEN ExRepD
THEN InstConcl [⌜e⌝]⋅
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. e1 : E@i
14. e2 : E@i
15. ldrs_uid : Id ⟶ ℤ@i
16. bnum : pv11_p1_Ballot_Num()@i
17. active : 𝔹@i
18. proposals : (ℤ × Cmd) List@i
19. s : ℤ@i
20. c : Cmd@i
21. (e1 <loc e2)@i
22. <bnum, active, proposals> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e1)@i
23. header(e1) = [propose] ∈ Name@i
24. has-es-info-type(es;e1;f;ℤ × Cmd)@i
25. <s, c> = msgval(e1) ∈ (ℤ × Cmd)@i
26. ¬↑(pv11_p1_in_domain(Cmd) s proposals)@i
27. e : {e:E| ¬↑first(e)}
28. e1 = pred(e) ∈ E
29. e ≤loc e2
30. e1 = pred(e) ∈ E
31. e ≤loc e2
⊢ <bnum, active, pv11_p1_add_if_new() pv11_p1_same_proposal(Cmd) <s, c> proposals> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)\000C(e)
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. e1 : E@i
14. e2 : E@i
15. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i
16. bnum : pv11\_p1\_Ballot\_Num()@i
17. active : \mBbbB{}@i
18. proposals : (\mBbbZ{} \mtimes{} Cmd) List@i
19. s : \mBbbZ{}@i
20. c : Cmd@i
21. (e1 <loc e2)@i
22. <bnum, active, proposals> \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e1)@i
23. header(e1) = [propose]@i
24. has-es-info-type(es;e1;f;\mBbbZ{} \mtimes{} Cmd)@i
25. <s, c> = msgval(e1)@i
26. \mneg{}\muparrow{}(pv11\_p1\_in\_domain(Cmd) s proposals)@i
\mvdash{} \mexists{}e:\{e:E| \mneg{}\muparrow{}first(e)\}
((e1 = pred(e))
\mwedge{} e \mleq{}loc e2
\mwedge{} <bnum, active, pv11\_p1\_add\_if\_new() pv11\_p1\_same\_proposal(Cmd) <s, c> proposals> \mmember{} pv11\_p1\_Lead\000CerState(Cmd;ldrs$_{uid}$;f)(e))
By
Latex:
((InstLemma `es-pred-exists-between2` [\mkleeneopen{}parm\{i'\}\mkleeneclose{};\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}e1\mkleeneclose{};\mkleeneopen{}e2\mkleeneclose{}]\mcdot{} THENA Auto)
THEN ExRepD
THEN InstConcl [\mkleeneopen{}e\mkleeneclose{}]\mcdot{}
THEN Auto)
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