Nuprl Lemma : pv11_p1_adopted_prior
∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀e:E. ∀ldrs_uid:Id ⟶ ℤ.
∀bnum:pv11_p1_Ballot_Num(). ∀active:𝔹. ∀proposals:(ℤ × Cmd) List. ∀pvals:(pv11_p1_Ballot_Num() × ℤ × Cmd) List.
∀b':pv11_p1_Ballot_Num(). ∀s:ℤ. ∀c,c':Cmd.
((¬↑first(e))
⇒ <bnum, active, proposals> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e)
⇒ (<s, c> ∈ proposals)
⇒ <bnum, pvals> ∈ pv11_p1_adopted'base(Cmd;f)(pred(e))
⇒ (<b', s, c'> ∈ pvals)
⇒ (↓∃b:pv11_p1_Ballot_Num(). ((<b, s, c> ∈ pvals) ∧ (↑(pv11_p1_leq_bnum(ldrs_uid) b' b)))))
Proof
Definitions occuring in Statement :
pv11_p1_LeaderState: pv11_p1_LeaderState(Cmd;ldrs_uid;mf),
pv11_p1_adopted'base: pv11_p1_adopted'base(Cmd;mf),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_leq_bnum: pv11_p1_leq_bnum(ldrs_uid),
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,
vatype: ValueAllType,
assert: ↑b,
bool: 𝔹,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
and: 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),
uiff: uiff(P;Q),
bool: 𝔹,
unit: Unit,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
pv11_p1_propose'base: pv11_p1_propose'base(Cmd;mf),
pv11_p1_adopted'base: pv11_p1_adopted'base(Cmd;mf),
encodes-msg-type: hdr encodes T,
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
top: Top,
strict4: strict4(F),
has-value: (a)↓,
has-es-info-type: has-es-info-type(es;e;f;T),
pv11_p1_preempted'base: pv11_p1_preempted'base(Cmd;mf),
single-valued-classrel: single-valued-classrel(es;X;T),
pv11_p1_when_adopted: pv11_p1_when_adopted(Cmd;ldrs_uid),
spreadn: spread3,
let: let,
pi1: fst(t),
pi2: snd(t),
cand: A c∧ B
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{}bnum:pv11\_p1\_Ballot\_Num(). \mforall{}active:\mBbbB{}. \mforall{}proposals:(\mBbbZ{} \mtimes{} Cmd) List. \mforall{}pvals:(pv11\_p1\_Ballot\_Num()
\mtimes{} \mBbbZ{}
\mtimes{} Cmd) List.
\mforall{}b':pv11\_p1\_Ballot\_Num(). \mforall{}s:\mBbbZ{}. \mforall{}c,c':Cmd.
((\mneg{}\muparrow{}first(e))
{}\mRightarrow{} <bnum, active, proposals> \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e)
{}\mRightarrow{} (<s, c> \mmember{} proposals)
{}\mRightarrow{} <bnum, pvals> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(pred(e))
{}\mRightarrow{} (<b', s, c'> \mmember{} pvals)
{}\mRightarrow{} (\mdownarrow{}\mexists{}b:pv11\_p1\_Ballot\_Num(). ((<b, s, c> \mmember{} pvals) \mwedge{} (\muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid\mbackslash{}ff\000C7d$) b' b)))))
Date html generated:
2016_05_17-PM-03_43_46
Last ObjectModification:
2016_01_18-AM-11_17_08
Theory : paxos!synod
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