Nuprl Lemma : pv11_p1_ldr_fun_mem_adopted
∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀e1,e2:E. ∀ldrs_uid:Id ⟶ ℤ.
∀pvals:(pv11_p1_Ballot_Num() × ℤ × Cmd) List. ∀s:ℤ. ∀c:Cmd.
((e1 <loc e2)
⇒ <fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)), pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e1)
⇒ ((↓(<s, c> ∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)))))
∨ (∃b:pv11_p1_Ballot_Num(). (↓(<b, s, c> ∈ pvals))))
⇒ (↑(pv11_p1_in_domain(Cmd) s (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2)))))))
Proof
Definitions occuring in Statement :
pv11_p1_LeaderStateFun: pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e),
pv11_p1_adopted'base: pv11_p1_adopted'base(Cmd;mf),
pv11_p1_in_domain: pv11_p1_in_domain(Cmd),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
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-E: E,
Id: Id,
l_member: (x ∈ l),
list: T List,
vatype: ValueAllType,
assert: ↑b,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
int: ℤ
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),
top: Top,
exists: ∃x:A. B[x],
or: P ∨ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
isl: isl(x),
outl: outl(x),
outr: outr(x),
assert: ↑b,
cand: A c∧ B,
spreadn: spread3,
pi1: fst(t),
sq_stable: SqStable(P)
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{}pvals:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List. \mforall{}s:\mBbbZ{}. \mforall{}c:Cm\000Cd.
((e1 <loc e2)
{}\mRightarrow{} <fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1)), pvals> \mmember{} pv11\_p1\_ad\000Copted'base(Cmd;f)(e1)
{}\mRightarrow{} ((\mdownarrow{}(<s, c> \mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1)))))
\mvee{} (\mexists{}b:pv11\_p1\_Ballot\_Num(). (\mdownarrow{}(<b, s, c> \mmember{} pvals))))
{}\mRightarrow{} (\muparrow{}(pv11\_p1\_in\_domain(Cmd) s (snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}\mbackslash{}ff2\000C4;f;es;e2)))))))
Date html generated:
2016_05_17-PM-03_41_01
Last ObjectModification:
2016_01_18-AM-11_16_57
Theory : paxos!synod
Home
Index