Nuprl 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``))))))
Proof
Definitions occuring in Statement :
pv11_p1_main: pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;mf),
pv11_p1_LeaderStateFun: pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e),
pv11_p1_scout_output: pv11_p1_scout_output(Cmd;accpts;mf),
pv11_p1_ScoutState: pv11_p1_ScoutState(Cmd;accpts;mf),
pv11_p1_commander_output: pv11_p1_commander_output(Cmd;accpts;reps;mf),
pv11_p1_CommanderState: pv11_p1_CommanderState(Cmd;accpts;mf),
pv11_p1_init_active: pv11_p1_init_active(),
pv11_p1_init_ballot_num: pv11_p1_init_ballot_num(),
pv11_p1_p2b'base: pv11_p1_p2b'base(Cmd;mf),
pv11_p1_p1b'base: pv11_p1_p1b'base(Cmd;mf),
pv11_p1_in_domain: pv11_p1_in_domain(Cmd),
pv11_p1_pmax: pv11_p1_pmax(Cmd;ldrs_uid),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_lt_bnum: pv11_p1_lt_bnum(ldrs_uid),
pv11_p1_upd_bnum: pv11_p1_upd_bnum(),
pv11_p1_is_bnum: pv11_p1_is_bnum(),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
msg-interface: Interface,
mk-msg: mk-msg(auth;hdr;val),
es-info-body: msgval(e),
has-es-info-type: has-es-info-type(es;e;f;T),
es-header: header(e),
Message: Message(f),
eclass2: (X o Y),
eclass1: (f o X),
no-classrel-in-interval: (no X between start and e),
no-prior-classrel: (no X prior to e),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
es-loc: loc(e),
es-E: E,
Id: Id,
name: Name,
l_member: (x ∈ l),
cons: [a / b],
nil: [],
list: T List,
valueall-type: valueall-type(T),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
squash: ↓T,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ─→ B[x],
pair: <a, b>,
product: x:A × B[x],
natural_number: $n,
int: ℤ,
token: "$token",
universe: Type,
equal: s = t ∈ T,
bag-member: x ↓∈ bs,
bag: bag(T)
Lemmas :
sq_stable__and,
equal_wf,
cons_wf_listp,
nil_wf,
listp_wf,
vatype_wf,
cons_wf,
list_wf,
equal-wf-T-base,
sq_stable__equal,
squash_wf,
int_seg_wf,
length_wf,
name_wf,
pv11_p1_headers_wf,
l_all_iff,
l_member_wf,
pv11_p1_headers_fun_wf,
cons_member,
equal-wf-base,
iff_weakening_equal,
classrel_wf,
msg-interface_wf,
pv11_p1_main_wf,
make-msg-interface_wf,
mk-msg_wf,
subtype_rel_weakening,
ext-eq_weakening,
bag-member_wf,
es-loc_wf,
event-ordering+_subtype,
or_wf,
no-prior-classrel_wf,
eclass2_wf,
eclass1_wf,
bag_wf,
pv11_p1_p1b'base_wf,
pv11_p1_scout_output_wf,
pv11_p1_init_ballot_num_wf,
pv11_p1_ScoutState_wf,
es-header_wf,
has-es-info-type_wf,
es-info-body_wf,
int_subtype_base,
not_wf,
bool_wf,
exists_wf,
es-E_wf,
es-le_wf,
assert_wf,
pv11_p1_LeaderStateFun_wf,
pv11_p1_in_domain_wf,
list-subtype-bag,
pv11_p1_pmax_wf,
pv11_p1_lt_bnum_wf,
no-classrel-in-interval_wf,
pv11_p1_p2b'base_wf,
pv11_p1_commander_output_wf,
pv11_p1_CommanderState_wf,
pv11_p1_is_bnum_wf,
pv11_p1_upd_bnum_wf,
pv11_p1_Ballot_Num_wf,
event-ordering+_wf,
Message_wf,
subtype_rel_dep_function,
pv11_p1_headers_type_wf,
Id_wf,
set_wf,
valueall-type_wf,
pv11_p1-ilf,
eo-forward_wf,
less_than_wf,
bag-size_wf,
pv11_p1_ScoutStateFun_wf,
nat_wf,
pv11_p1_threshold_wf,
make-Msg_wf,
member-eo-forward-E,
eo-forward-header,
eo-forward-has-es-info-type,
equal-wf-base-T,
pv11_p1_CommanderStateFun_wf,
es-first_wf2,
pv11_p1_AcceptorStateFun_wf,
msg-header_wf,
hdrmkmsg_lemma,
hdrmakeMsg_lemma,
sq_stable__no_repeats,
true_wf,
false_wf,
and_wf,
make-Msg-as-mk-msg,
mk-msg-equal,
bfalse_wf,
null_nil_lemma,
btrue_wf,
reduce_tl_cons_lemma,
tl_wf,
null_wf3,
subtype_rel_list,
top_wf,
null_cons_lemma,
btrue_neq_bfalse,
eo-forward-info-body,
eo-forward-loc,
member_wf,
sq_stable__bag-member,
assert-name_eq
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))))))))))
Date html generated:
2015_07_23-PM-04_42_54
Last ObjectModification:
2015_04_08-AM-01_26_58
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