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)
Definitions unfolded in proof :
vatype: ValueAllType,
all: ∀x:A. B[x],
implies: P ⇒ Q,
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
l_all: (∀x∈L.P[x]),
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
listp: A List+,
name: Name,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
length: ||as||,
list_ind: list_ind,
pv11_p1_headers: pv11_p1_headers(),
cons: [a / b],
nil: [],
it: ⋅,
true: True,
select: L[n],
subtract: n - m,
uimplies: b supposing a,
guard: {T},
rev_implies: P ⇐ Q,
pv11_p1_headers_fun: pv11_p1_headers_fun(Cmd),
name_eq: name_eq(x;y),
name-deq: NameDeq,
list-deq: list-deq(eq),
band: p ∧b q,
ifthenelse: if b then t else f fi ,
atom-deq: AtomDeq,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
null: null(as),
pv11_p1_message-constraint: pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; mf; es),
eo-msg-interface-constraint: eo-msg-interface-constraint(es;X;hdrs;f),
pv11_p1_adopted'base: pv11_p1_adopted'base(Cmd;mf),
uiff: uiff(P;Q),
es-header: header(e),
pv11_p1_headers_no_inputs: pv11_p1_headers_no_inputs(),
top: Top,
or: P ∨ Q,
exists: ∃x:A. B[x],
make-msg-interface: make-msg-interface(i;l;m),
has-es-info-type: has-es-info-type(es;e;f;T),
cand: A c∧ B,
es-locl: (e <loc e'),
pv11_p1_preempted'base: pv11_p1_preempted'base(Cmd;mf),
encodes-msg-type: hdr encodes T,
rev_uimplies: rev_uimplies(P;Q)
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:
2016_05_17-PM-03_41_59
Last ObjectModification:
2016_01_18-AM-11_16_46
Theory : paxos!synod
Home
Index