Nuprl Lemma : pv11_p1_A2
∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀e:E. ∀ldrs_uid:Id ⟶ ℤ.
∀bnum1,bnum2:pv11_p1_Ballot_Num(). ∀accepted1,accepted2:(pv11_p1_Ballot_Num() × ℤ × Cmd) List. ∀b:pv11_p1_Ballot_Num().
∀pv:ℤ × Cmd. ∀loc:Id.
(Inj(Id;ℤ;ldrs_uid)
⇒ (¬↑first(e))
⇒ <bnum2, accepted2> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e)
⇒ <bnum1, accepted1> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(pred(e))
⇒ (<b, pv> ∈ accepted2)
⇒ (¬(<b, pv> ∈ accepted1))
⇒ (b = bnum2 ∈ pv11_p1_Ballot_Num()))
Proof
Definitions occuring in Statement :
pv11_p1_AcceptorState: pv11_p1_AcceptorState(Cmd;ldrs_uid;mf),
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-first: first(e),
es-pred: pred(e),
es-E: E,
Id: Id,
l_member: (x ∈ l),
list: T List,
inject: Inj(A;B;f),
vatype: ValueAllType,
assert: ↑b,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
int: ℤ,
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),
iseg: l1 ≤ l2,
exists: ∃x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
bor: p ∨bq,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
pv11_p1_p2a'base: pv11_p1_p2a'base(Cmd;mf),
encodes-msg-type: hdr encodes T,
pv11_p1_p1a'base: pv11_p1_p1a'base(Cmd;mf),
pv11_p1_on_p1a: pv11_p1_on_p1a(Cmd;ldrs_uid),
msg-has-type: msg-has-type(m;f;T),
msg-type: msg-type(msg;f),
es-header: header(e),
has-es-info-type: has-es-info-type(es;e;f;T),
top: Top,
pi1: fst(t),
pi2: snd(t),
pv11_p1_on_p2a: pv11_p1_on_p2a(Cmd;ldrs_uid),
let: let,
pv11_p1_add_if_new: pv11_p1_add_if_new()
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)). \mforall{}e:E. \mforall{}ldrs$_\mbackslash{}ff7\000Cbuid}$:Id {}\mrightarrow{} \mBbbZ{}.
\mforall{}bnum1,bnum2:pv11\_p1\_Ballot\_Num(). \mforall{}accepted1,accepted2:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List.
\mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}pv:\mBbbZ{} \mtimes{} Cmd. \mforall{}loc:Id.
(Inj(Id;\mBbbZ{};ldrs$_{uid}$)
{}\mRightarrow{} (\mneg{}\muparrow{}first(e))
{}\mRightarrow{} <bnum2, accepted2> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e)
{}\mRightarrow{} <bnum1, accepted1> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(pred(e))
{}\mRightarrow{} (<b, pv> \mmember{} accepted2)
{}\mRightarrow{} (\mneg{}(<b, pv> \mmember{} accepted1))
{}\mRightarrow{} (b = bnum2))
Date html generated:
2016_05_17-PM-03_59_06
Last ObjectModification:
2016_01_18-AM-11_17_18
Theory : paxos!synod
Home
Index