Nuprl Lemma : pv11_p1-agreement
∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀e1,e2:E. ∀accpts,ldrs:bag(Id).
∀ldrs_uid:Id ⟶ ℤ. ∀reps:bag(Id). ∀i1,i2:Id. ∀s:ℤ. ∀c1,c2:Cmd.
(pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)
⇒ Inj(Id;ℤ;ldrs_uid)
⇒ pv11_p1_decision'send(Cmd;f) i1 <s, c1> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e1)
⇒ pv11_p1_decision'send(Cmd;f) i2 <s, c2> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e2)
⇒ (c1 = c2 ∈ Cmd))
Proof
Definitions occuring in Statement :
pv11_p1_decision'send: pv11_p1_decision'send(Cmd;f),
pv11_p1_message-constraint: pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; mf; es),
pv11_p1_main: pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;mf),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
msg-interface: Interface,
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-E: E,
Id: Id,
inject: Inj(A;B;f),
vatype: ValueAllType,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
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_decision'send: pv11_p1_decision'send(Cmd;f),
top: Top,
mk-msg-interface: mk-msg-interface(l;m),
make-msg-interface: make-msg-interface(i;l;m),
exists: ∃x:A. B[x],
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),
sq_type: SQType(T),
assert: ↑b,
deq-member: x ∈b L,
reduce: reduce(f;k;as),
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
strict4: strict4(F),
has-value: (a)↓,
or: P ∨ Q,
pv11_p1_headers_no_inputs: pv11_p1_headers_no_inputs(),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bor: p ∨bq,
has-es-info-type: has-es-info-type(es;e;f;T),
es-info-body: msgval(e),
es-info-auth: es-info-auth(es;e),
pi2: snd(t),
pi1: fst(t),
sq_stable: SqStable(P),
pv11_p1_decision: pv11_p1_decision{i:l}(Cmd;f;accpts;ldrs;ldrs_uid;reps;es;e;k;v),
pv11_p1_pvalue: pv11_p1_pvalue{i:l}(Cmd;ldrs_uid;accpts;es;e;x;f),
pv11_p1_p2a'base: pv11_p1_p2a'base(Cmd;mf),
encodes-msg-type: hdr encodes T,
uiff: uiff(P;Q),
cand: A c∧ B,
equal-info-body: v = body(e),
es-info-type: es-info-type(es;e;f),
msg-type: msg-type(msg;f),
es-header: header(e)
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)). \mforall{}e1,e2:E.
\mforall{}accpts,ldrs:bag(Id). \mforall{}ldrs$_{uid}$:Id {}\mrightarrow{} \mBbbZ{}. \mforall{}reps:bag(Id). \mforall{}i1,i2:Id. \mforall{}s:\mBbbZ{}. \mforall{}c1\000C,c2:Cmd.
(pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\mbackslash{}\000Cff24; reps; f; es)
{}\mRightarrow{} Inj(Id;\mBbbZ{};ldrs$_{uid}$)
{}\mRightarrow{} pv11\_p1\_decision'send(Cmd;f) i1 <s, c1> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid\mbackslash{}f\000Cf7d$;reps;f)(e1)
{}\mRightarrow{} pv11\_p1\_decision'send(Cmd;f) i2 <s, c2> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid\mbackslash{}f\000Cf7d$;reps;f)(e2)
{}\mRightarrow{} (c1 = c2))
Date html generated:
2016_05_17-PM-04_19_23
Last ObjectModification:
2016_01_18-AM-11_18_04
Theory : paxos!synod
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