Nuprl Lemma : pv11_p1_ldr_state_adopted_pred
∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀e1,e2:E. ∀ldrs_uid:Id ⟶ ℤ.
∀bnum:pv11_p1_Ballot_Num(). ∀active:𝔹. ∀proposals:(ℤ × Cmd) List. ∀pvals:(pv11_p1_Ballot_Num() × ℤ × Cmd) List.
((e1 <loc e2)
⇒ <bnum, active, proposals> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e1)
⇒ (header(e1) = ``pv11_p1 adopted`` ∈ Name)
⇒ (<bnum, pvals> = msgval(e1) ∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
⇒ (∃e:{e:E| ¬↑first(e)}
((e1 = pred(e) ∈ E)
∧ e ≤loc e2
∧ <bnum, tt, pv11_p1_update_proposals(Cmd) proposals (pv11_p1_pmax(Cmd;ldrs_uid) pvals)> ∈
pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e))))
Proof
Definitions occuring in Statement :
pv11_p1_LeaderState: pv11_p1_LeaderState(Cmd;ldrs_uid;mf),
pv11_p1_update_proposals: pv11_p1_update_proposals(Cmd),
pv11_p1_pmax: pv11_p1_pmax(Cmd;ldrs_uid),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
es-info-body: msgval(e),
es-header: header(e),
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
es-locl: (e <loc e'),
es-first: first(e),
es-pred: pred(e),
es-E: E,
Id: Id,
name: Name,
cons: [a / b],
nil: [],
list: T List,
vatype: ValueAllType,
assert: ↑b,
btrue: tt,
bool: 𝔹,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: 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],
int: ℤ,
token: "$token",
equal: s = t ∈ T
Definitions unfolded in proof :
vatype: ValueAllType,
all: ∀x:A. B[x],
implies: P ⇒ Q,
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
l_all: (∀x∈L.P[x]),
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
listp: A List+,
name: Name,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
length: ||as||,
list_ind: list_ind,
pv11_p1_headers: pv11_p1_headers(),
cons: [a / b],
nil: [],
it: ⋅,
true: True,
select: L[n],
subtract: n - m,
uimplies: b supposing a,
guard: {T},
rev_implies: P ⇐ Q,
pv11_p1_headers_fun: pv11_p1_headers_fun(Cmd),
name_eq: name_eq(x;y),
name-deq: NameDeq,
list-deq: list-deq(eq),
band: p ∧b q,
ifthenelse: if b then t else f fi ,
atom-deq: AtomDeq,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
null: null(as),
exists: ∃x:A. B[x],
cand: A c∧ B,
es-info-type: es-info-type(es;e;f),
msg-type: msg-type(msg;f),
es-header: header(e),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
bool: 𝔹,
unit: Unit,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
pv11_p1_propose'base: pv11_p1_propose'base(Cmd;mf),
top: Top,
pv11_p1_adopted'base: pv11_p1_adopted'base(Cmd;mf),
encodes-msg-type: hdr encodes T,
has-es-info-type: has-es-info-type(es;e;f;T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
pv11_p1_when_adopted: pv11_p1_when_adopted(Cmd;ldrs_uid),
spreadn: spread3,
let: let,
pv11_p1_preempted'base: pv11_p1_preempted'base(Cmd;mf)
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)). \mforall{}e1,e2:E.
\mforall{}ldrs$_{uid}$:Id {}\mrightarrow{} \mBbbZ{}. \mforall{}bnum:pv11\_p1\_Ballot\_Num(). \mforall{}active:\mBbbB{}. \mforall{}proposals:(\mBbbZ{} \mtimes{} Cm\000Cd) List.
\mforall{}pvals:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List.
((e1 <loc e2)
{}\mRightarrow{} <bnum, active, proposals> \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e1)
{}\mRightarrow{} (header(e1) = ``pv11\_p1 adopted``)
{}\mRightarrow{} (<bnum, pvals> = msgval(e1))
{}\mRightarrow{} (\mexists{}e:\{e:E| \mneg{}\muparrow{}first(e)\}
((e1 = pred(e))
\mwedge{} e \mleq{}loc e2
\mwedge{} <bnum, tt, pv11\_p1\_update\_proposals(Cmd) proposals (pv11\_p1\_pmax(Cmd;ldrs$_{uid\000C}$) pvals)> \mmember{}
pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e))))
Date html generated:
2016_05_17-PM-03_44_51
Last ObjectModification:
2016_01_18-AM-11_17_13
Theory : paxos!synod
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