Step
*
of Lemma
pv11_p1-preempted
∀[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()].
(<d, i, mk-msg(auth;``pv11_p1 preempted``;p)> ∈ 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()))
∧ auth = pv11_p1_init_active()
∧ (p = (fst(snd(snd(msgval(e))))) ∈ (mf ``pv11_p1 preempted``)))
∨ (∃z4:Cmd
∃e':{e':E| e' ≤loc e }
((header(e) = ``pv11_p1 p2b`` ∈ Name)
∧ has-es-info-type(es;e;mf;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
∧ (d = 0 ∈ ℤ)
∧ (¬((fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e))))) ∈ pv11_p1_Ballot_Num()))
∧ (i = loc(e) ∈ Id)
∧ auth = pv11_p1_init_active()
∧ (p = (snd(snd(snd(msgval(e))))) ∈ (mf ``pv11_p1 preempted``))
∧ ((((header(e') = [propose] ∈ Name) ∧ has-es-info-type(es;e';mf;ℤ × Cmd))
∧ ((↑(fst(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e')))))
∧ (¬↑(pv11_p1_in_domain(Cmd) (fst(msgval(e')))
(snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e')))))))
∧ ((fst(snd(msgval(e)))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e'))) ∈ pv11_p1_Ballot_Num())
∧ (<fst(snd(snd(msgval(e)))), z4> = msgval(e') ∈ (ℤ × Cmd)))
∨ (((header(e') = ``pv11_p1 adopted`` ∈ Name)
∧ has-es-info-type(es;e';mf;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
∧ ((fst(msgval(e'))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e'))) ∈ pv11_p1_Ballot_Num())
∧ ((<fst(snd(snd(msgval(e)))), z4> ↓∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e')))
∧ (¬(∃p2:Cmd. (<fst(snd(snd(msgval(e)))), p2> ∈ pv11_p1_pmax(Cmd;ldrs_uid) (snd(msgval(e')))))))
∨ (∃v2:pv11_p1_Ballot_Num()
(<v2, fst(snd(snd(msgval(e)))), z4> ↓∈ snd(msgval(e'))
∧ (¬(∃z5:pv11_p1_Ballot_Num()
∃z8:Cmd. ((↑(v2 < z5)) ∧ (<z5, fst(snd(snd(msgval(e)))), z8> ∈ snd(msgval(e')))))))))
∧ ((fst(snd(msgval(e)))) = (fst(msgval(e'))) ∈ pv11_p1_Ballot_Num())))
∧ (no ((pv11_p1_commander_output(Cmd;accpts;reps;mf) <fst(snd(msgval(e))), fst(snd(snd(msgval(e)))), z4> \000Co pv11_p1_p2b'base(Cmd;mf)) o
pv11_p1_CommanderState(Cmd;accpts;mf) (fst(snd(msgval(e)))) (fst(snd(snd(msgval(e)))))) between e'
and e))))
∨ (∃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()))
∧ auth = pv11_p1_init_active()
∧ (p = (fst(snd(snd(msgval(e))))) ∈ (mf ``pv11_p1 preempted``))))))
BY
{ ProveILF_instance "preempted""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()].
(<d, i, mk-msg(auth;``pv11\_p1 preempted``;p)> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{ui\000Cd}$;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{} (\mneg{}((pv11\_p1\_init\_ballot\_num() loc(e)) = (fst(snd(snd(msgval(e)))))))
\mwedge{} auth = pv11\_p1\_init\_active()
\mwedge{} (p = (fst(snd(snd(msgval(e)))))))
\mvee{} (\mexists{}z4:Cmd
\mexists{}e':\{e':E| e' \mleq{}loc e \}
((header(e) = ``pv11\_p1 p2b``)
\mwedge{} has-es-info-type(es;e;mf;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num())
\mwedge{} (d = 0)
\mwedge{} (\mneg{}((fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e)))))))
\mwedge{} (i = loc(e))
\mwedge{} auth = pv11\_p1\_init\_active()
\mwedge{} (p = (snd(snd(snd(msgval(e))))))
\mwedge{} ((((header(e') = [propose]) \mwedge{} has-es-info-type(es;e';mf;\mBbbZ{} \mtimes{} Cmd))
\mwedge{} ((\muparrow{}(fst(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;mf;es;e')\000C))))
\mwedge{} (\mneg{}\muparrow{}(pv11\_p1\_in\_domain(Cmd) (fst(msgval(e')))
(snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;mf;es;\000Ce')))))))
\mwedge{} ((fst(snd(msgval(e)))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid\mbackslash{}f\000Cf7d$;mf;es;e'))))
\mwedge{} (<fst(snd(snd(msgval(e)))), z4> = msgval(e')))
\mvee{} (((header(e') = ``pv11\_p1 adopted``)
\mwedge{} has-es-info-type(es;e';mf;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num()
\mtimes{} \mBbbZ{}
\mtimes{} Cmd) List)))
\mwedge{} ((fst(msgval(e'))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid\mbackslash{}ff7\000Cd$;mf;es;e'))))
\mwedge{} ((<fst(snd(snd(msgval(e))))
, z4
> \mdownarrow{}\mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;mf;e\000Cs;e')))
\mwedge{} (\mneg{}(\mexists{}p2:Cmd
(<fst(snd(snd(msgval(e)))), p2> \mmember{} pv11\_p1\_pmax(Cmd;ldrs$_{uid\000C}$)
(snd(msgval(e')))))))
\mvee{} (\mexists{}v2:pv11\_p1\_Ballot\_Num()
(<v2, fst(snd(snd(msgval(e)))), z4> \mdownarrow{}\mmember{} snd(msgval(e'))
\mwedge{} (\mneg{}(\mexists{}z5:pv11\_p1\_Ballot\_Num()
\mexists{}z8:Cmd
((\muparrow{}(v2 < z5))
\mwedge{} (<z5, fst(snd(snd(msgval(e)))), z8> \mmember{} snd(msgval(e')))))))))
\mwedge{} ((fst(snd(msgval(e)))) = (fst(msgval(e'))))))
\mwedge{} (no ((pv11\_p1\_commander\_output(Cmd;accpts;reps;mf)
<fst(snd(msgval(e))), fst(snd(snd(msgval(e)))), z4> o
pv11\_p1\_p2b'base(Cmd;mf)) o pv11\_p1\_CommanderState(Cmd;accpts;mf)
(fst(snd(msgval(e))))
(fst(snd(snd(msgval(e)))))) between e' and 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{} (\mneg{}((pv11\_p1\_upd\_bnum() msgval(e') loc(e')) = (fst(snd(snd(msgval(e)))))))
\mwedge{} auth = pv11\_p1\_init\_active()
\mwedge{} (p = (fst(snd(snd(msgval(e))))))))))
By
Latex:
ProveILF\_instance "preempted""pv11\_p1"
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