Nuprl Lemma : pv11_p1_adopted_from_init_or_preempted
∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀e:E. ∀accpts,ldrs:bag(Id).
∀ldrs_uid:Id ─→ ℤ. ∀reps:bag(Id). ∀bnum:pv11_p1_Ballot_Num(). ∀pvals:(pv11_p1_Ballot_Num() × ℤ × Cmd) List.
(pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)
⇒ <bnum, pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e)
⇒ (↓(bnum = (pv11_p1_init_ballot_num() loc(e)) ∈ pv11_p1_Ballot_Num())
∨ (∃e':E
∃bnum1,bnum2:pv11_p1_Ballot_Num()
∃b:𝔹
∃proposals:(ℤ × Cmd) List
((e' <loc e)
∧ bnum1 ∈ pv11_p1_preempted'base(Cmd;f)(e')
∧ <bnum2, b, proposals> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e')
∧ (↑(bnum2 < bnum1))
∧ (bnum = (pv11_p1_upd_bnum() bnum1 loc(e)) ∈ pv11_p1_Ballot_Num())))))
Proof
Definitions occuring in Statement :
pv11_p1_message-constraint: pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; mf; es),
pv11_p1_LeaderState: pv11_p1_LeaderState(Cmd;ldrs_uid;mf),
pv11_p1_init_ballot_num: pv11_p1_init_ballot_num(),
pv11_p1_adopted'base: pv11_p1_adopted'base(Cmd;mf),
pv11_p1_preempted'base: pv11_p1_preempted'base(Cmd;mf),
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_Ballot_Num: pv11_p1_Ballot_Num(),
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-locl: (e <loc e'),
es-loc: loc(e),
es-E: E,
Id: Id,
list: T List,
vatype: ValueAllType,
assert: ↑b,
bool: 𝔹,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ─→ B[x],
pair: <a, b>,
product: x:A × B[x],
int: ℤ,
equal: s = t ∈ T,
bag: bag(T)
Lemmas :
int_seg_wf,
length_wf,
name_wf,
pv11_p1_headers_wf,
l_all_iff,
l_member_wf,
equal_wf,
pv11_p1_headers_fun_wf,
cons_wf_listp,
nil_wf,
listp_wf,
cons_member,
cons_wf,
equal-wf-base,
iff_weakening_equal,
base-noloc-classrel-make-Msg2,
hdrmkmsg_lemma,
msg-header_wf,
pv11_p1_headers_no_inputs_wf,
squash_wf,
exists_wf,
es-E_wf,
event-ordering+_subtype,
es-causl_wf,
classrel_wf,
msg-interface_wf,
pv11_p1_main_wf,
make-msg-interface_wf,
es-loc_wf,
pv11_p1_adopted'base_wf,
pv11_p1_message-constraint_wf,
list_wf,
pv11_p1_Ballot_Num_wf,
bag_wf,
Id_wf,
event-ordering+_wf,
Message_wf,
subtype_rel_dep_function,
vatype_wf,
pv11_p1_headers_type_wf,
set_wf,
valueall-type_wf,
msg-authentic_wf,
es-info_wf,
pv11_p1-adopted,
pv11_p1_init_ballot_num_wf,
bool_wf,
es-locl_wf,
pv11_p1_preempted'base_wf,
pv11_p1_LeaderState_wf,
assert_wf,
pv11_p1_lt_bnum_wf,
pv11_p1_upd_bnum_wf,
es-info-body_wf,
pv11_p1_LeaderStateFun_wf,
es-le-loc,
es-causl_transitivity1,
es-causle_weakening_locl,
and_wf,
base-noloc-classrel,
subtype_rel_weakening,
ext-eq_weakening,
pv11_p1_LeaderState-classrel,
pair-eta,
pair_eta_rw
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)). \mforall{}e:E.
\mforall{}accpts,ldrs:bag(Id). \mforall{}ldrs$_{uid}$:Id {}\mrightarrow{} \mBbbZ{}. \mforall{}reps:bag(Id). \mforall{}bnum:pv11\_p1\_Ballot\000C\_Num().
\mforall{}pvals:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List.
(pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\mbackslash{}\000Cff24; reps; f; es)
{}\mRightarrow{} <bnum, pvals> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e)
{}\mRightarrow{} (\mdownarrow{}(bnum = (pv11\_p1\_init\_ballot\_num() loc(e)))
\mvee{} (\mexists{}e':E
\mexists{}bnum1,bnum2:pv11\_p1\_Ballot\_Num()
\mexists{}b:\mBbbB{}
\mexists{}proposals:(\mBbbZ{} \mtimes{} Cmd) List
((e' <loc e)
\mwedge{} bnum1 \mmember{} pv11\_p1\_preempted'base(Cmd;f)(e')
\mwedge{} <bnum2, b, proposals> \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(\000Ce')
\mwedge{} (\muparrow{}(bnum2 < bnum1))
\mwedge{} (bnum = (pv11\_p1\_upd\_bnum() bnum1 loc(e)))))))
Date html generated:
2015_07_23-PM-04_47_38
Last ObjectModification:
2015_02_04-AM-08_20_42
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