Step
*
of Lemma
pv11_p1-adopted
∀[Cmd:{T:Type| valueall-type(T)} ]. ∀[accpts,ldrs:bag(Id)]. ∀[ldrs_uid:Id ─→ ℤ]. ∀[reps:bag(Id)].
∀[mf:pv11_p1_headers_type{i:l}(Cmd)]. ∀[es:EO+(Message(mf))]. ∀[e:E]. ∀[d:ℤ]. ∀[i:Id]. ∀[auth:𝔹].
∀[p:pv11_p1_Ballot_Num()]. ∀[L:(pv11_p1_Ballot_Num() × ℤ × Cmd) List].
(<d, i, mk-msg(auth;``pv11_p1 adopted``;<p, L>)> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;mf)(e)
⇐⇒ loc(e) ↓∈ ldrs
∧ (↓((no ((pv11_p1_scout_output(Cmd;accpts;mf) (pv11_p1_init_ballot_num() loc(e)) o pv11_p1_p1b'base(Cmd;mf)) o
pv11_p1_ScoutState(Cmd;accpts;mf) (pv11_p1_init_ballot_num() loc(e))) prior to e)
∧ ((header(e) = ``pv11_p1 p1b`` ∈ Name)
∧ has-es-info-type(es;e;mf;Id
× pv11_p1_Ballot_Num()
× pv11_p1_Ballot_Num()
× ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
∧ ((pv11_p1_init_ballot_num() loc(e)) = (fst(snd(msgval(e)))) ∈ pv11_p1_Ballot_Num())
∧ (d = 0 ∈ ℤ)
∧ (i = loc(e) ∈ Id)
∧ ((pv11_p1_init_ballot_num() loc(e)) = (fst(snd(snd(msgval(e))))) ∈ pv11_p1_Ballot_Num())
∧ #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;mf;pv11_p1_init_ballot_num()
loc(e);es;e))) < pv11_p1_threshold(accpts)
∧ auth = pv11_p1_init_active()
∧ (p = (pv11_p1_init_ballot_num() loc(e)) ∈ pv11_p1_Ballot_Num())
∧ (L
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;mf;pv11_p1_init_ballot_num() loc(e);es;e)))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
∨ (∃e':{e':E| e' ≤loc e }
((((header(e') = ``pv11_p1 preempted`` ∈ Name) ∧ has-es-info-type(es;e';mf;pv11_p1_Ballot_Num()))
∧ (↑(pv11_p1_is_bnum() msgval(e')))
∧ (↑(fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e')) < msgval(e'))))
∧ (no ((pv11_p1_scout_output(Cmd;accpts;mf) (pv11_p1_upd_bnum() msgval(e') loc(e')) o
pv11_p1_p1b'base(Cmd;mf)) o pv11_p1_ScoutState(Cmd;accpts;mf)
(pv11_p1_upd_bnum() msgval(e') loc(e'))) between e' and e)
∧ ((header(e) = ``pv11_p1 p1b`` ∈ Name)
∧ has-es-info-type(es;e;mf;Id
× pv11_p1_Ballot_Num()
× pv11_p1_Ballot_Num()
× ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
∧ ((pv11_p1_upd_bnum() msgval(e') loc(e')) = (fst(snd(msgval(e)))) ∈ pv11_p1_Ballot_Num())
∧ (d = 0 ∈ ℤ)
∧ (i = loc(e) ∈ Id)
∧ ((pv11_p1_upd_bnum() msgval(e') loc(e')) = (fst(snd(snd(msgval(e))))) ∈ pv11_p1_Ballot_Num())
∧ #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;mf;pv11_p1_upd_bnum() msgval(e')
loc(e');es.e';e))) < pv11_p1_threshold(accpts)
∧ auth = pv11_p1_init_active()
∧ (p = (pv11_p1_upd_bnum() msgval(e') loc(e')) ∈ pv11_p1_Ballot_Num())
∧ (L
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;mf;pv11_p1_upd_bnum() msgval(e') loc(e');es.e';e)))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))))))
BY
{ ProveILF_instance "adopted""pv11_p1" }
Latex:
Latex:
\mforall{}[Cmd:\{T:Type| valueall-type(T)\} ]. \mforall{}[accpts,ldrs:bag(Id)]. \mforall{}[ldrs$_{uid}$:Id {}\mrightarrow{}\000C \mBbbZ{}]. \mforall{}[reps:bag(Id)].
\mforall{}[mf:pv11\_p1\_headers\_type\{i:l\}(Cmd)]. \mforall{}[es:EO+(Message(mf))]. \mforall{}[e:E]. \mforall{}[d:\mBbbZ{}]. \mforall{}[i:Id]. \mforall{}[auth:\mBbbB{}].
\mforall{}[p:pv11\_p1\_Ballot\_Num()]. \mforall{}[L:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List].
(<d, i, mk-msg(auth;``pv11\_p1 adopted``;<p, L>)> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_\mbackslash{}ff7\000Cbuid}$;reps;mf)(e)
\mLeftarrow{}{}\mRightarrow{} loc(e) \mdownarrow{}\mmember{} ldrs
\mwedge{} (\mdownarrow{}((no ((pv11\_p1\_scout\_output(Cmd;accpts;mf) (pv11\_p1\_init\_ballot\_num() loc(e)) o
pv11\_p1\_p1b'base(Cmd;mf)) o pv11\_p1\_ScoutState(Cmd;accpts;mf)
(pv11\_p1\_init\_ballot\_num() loc(e))) prior to e)
\mwedge{} ((header(e) = ``pv11\_p1 p1b``)
\mwedge{} has-es-info-type(es;e;mf;Id
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)))
\mwedge{} ((pv11\_p1\_init\_ballot\_num() loc(e)) = (fst(snd(msgval(e)))))
\mwedge{} (d = 0)
\mwedge{} (i = loc(e))
\mwedge{} ((pv11\_p1\_init\_ballot\_num() loc(e)) = (fst(snd(snd(msgval(e))))))
\mwedge{} \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;mf;pv11\_p1\_init\_ballot\_num()
loc(e);es;e))) < pv11\_p1\_threshold(accpts)
\mwedge{} auth = pv11\_p1\_init\_active()
\mwedge{} (p = (pv11\_p1\_init\_ballot\_num() loc(e)))
\mwedge{} (L = (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;mf;pv11\_p1\_init\_ballot\_num() loc(e);es;e)))))
\mvee{} (\mexists{}e':\{e':E| e' \mleq{}loc e \}
((((header(e') = ``pv11\_p1 preempted``)
\mwedge{} has-es-info-type(es;e';mf;pv11\_p1\_Ballot\_Num()))
\mwedge{} (\muparrow{}(pv11\_p1\_is\_bnum() msgval(e')))
\mwedge{} (\muparrow{}(fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;mf;es;e')) < ms\000Cgval(e'))))
\mwedge{} (no ((pv11\_p1\_scout\_output(Cmd;accpts;mf) (pv11\_p1\_upd\_bnum() msgval(e') loc(e')) o
pv11\_p1\_p1b'base(Cmd;mf)) o pv11\_p1\_ScoutState(Cmd;accpts;mf)
(pv11\_p1\_upd\_bnum() msgval(e') loc(e'))) between e'
and e)
\mwedge{} ((header(e) = ``pv11\_p1 p1b``)
\mwedge{} has-es-info-type(es;e;mf;Id
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)))
\mwedge{} ((pv11\_p1\_upd\_bnum() msgval(e') loc(e')) = (fst(snd(msgval(e)))))
\mwedge{} (d = 0)
\mwedge{} (i = loc(e))
\mwedge{} ((pv11\_p1\_upd\_bnum() msgval(e') loc(e')) = (fst(snd(snd(msgval(e))))))
\mwedge{} \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;mf;pv11\_p1\_upd\_bnum() msgval(e')
loc(e');es.e';e))) < ...
\mwedge{} auth = pv11\_p1\_init\_active()
\mwedge{} (p = (pv11\_p1\_upd\_bnum() msgval(e') loc(e')))
\mwedge{} (L
= (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;mf;pv11\_p1\_upd\_bnum() msgval(e')
loc(e');es.e';e))))))))
By
Latex:
ProveILF\_instance "adopted""pv11\_p1"
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