Nuprl Lemma : pv11_p1_consistency_lemma
∀Cmd:{T:Type| valueall-type(T)} . ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀accpts,ldrs:bag(Id).
∀ldrs_uid:Id ⟶ ℤ. ∀reps:bag(Id).
(pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)
⇒ Inj(Id;ℤ;ldrs_uid)
⇒ (∀e:E. ∀b:pv11_p1_Ballot_Num(). ∀k:ℤ. ∀v:Cmd.
(pv11_p1_pvalue{i:l}(Cmd;ldrs_uid;accpts;es;e;<b, k, v>;f)
⇒ (∀e':E. ∀w:Cmd.
(pv11_p1_decision{i:l}(Cmd;f;accpts;ldrs;ldrs_uid;reps;es;e';k;w)
⇒ has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) (snd(snd(snd(msgval(e'))))) b))
⇒ (w = v ∈ Cmd))))))
Proof
Definitions occuring in Statement :
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_message-constraint: pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; mf; es),
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(),
es-info-body: msgval(e),
has-es-info-type: has-es-info-type(es;e;f;T),
Message: Message(f),
event-ordering+: EO+(Info),
es-E: E,
Id: Id,
inject: Inj(A;B;f),
valueall-type: valueall-type(T),
assert: ↑b,
pi2: snd(t),
all: ∀x:A. B[x],
implies: 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,
universe: Type,
equal: s = t ∈ T,
bag: bag(T)
Definitions unfolded in proof :
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,
vatype: ValueAllType,
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),
has-es-info-type: has-es-info-type(es;e;f;T),
pi2: snd(t),
strongwellfounded: SWellFounded(R[x; y]),
exists: ∃x:A. B[x],
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
decidable: Dec(P),
or: P ∨ Q,
es-causle: e c≤ e',
pv11_p1_pvalue: pv11_p1_pvalue{i:l}(Cmd;ldrs_uid;accpts;es;e;x;f),
cand: A c∧ B,
pv11_p1_from-p2a: pv11_p1_from-p2a{i:l}(Cmd;es;e;x),
es-info-type: es-info-type(es;e;f),
msg-type: msg-type(msg;f),
es-header: header(e),
sq_type: SQType(T),
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),
pv11_p1_headers_no_inputs: pv11_p1_headers_no_inputs(),
es-info-body: msgval(e),
es-info-auth: es-info-auth(es;e),
pi1: fst(t),
make-msg-interface: make-msg-interface(i;l;m),
pv11_p1_p2a'send: pv11_p1_p2a'send(Cmd;f),
mk-msg: mk-msg(auth;hdr;val),
make-Msg: make-Msg(hdr;val),
mk-msg-interface: mk-msg-interface(l;m),
pv11_p1_init_active: pv11_p1_init_active(),
pv11_p1_decision: pv11_p1_decision{i:l}(Cmd;f;accpts;ldrs;ldrs_uid;reps;es;e;k;v),
pv11_p1_decision'send: pv11_p1_decision'send(Cmd;f),
uiff: uiff(P;Q)
Latex:
\mforall{}Cmd:\{T:Type| valueall-type(T)\} . \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)).
\mforall{}accpts,ldrs:bag(Id). \mforall{}ldrs$_{uid}$:Id {}\mrightarrow{} \mBbbZ{}. \mforall{}reps:bag(Id).
(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}$)
{}\mRightarrow{} (\mforall{}e:E. \mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}k:\mBbbZ{}. \mforall{}v:Cmd.
(pv11\_p1\_pvalue\{i:l\}(Cmd;ldrs$_{uid}$;accpts;es;e;<b, k, v>f)
{}\mRightarrow{} (\mforall{}e':E. \mforall{}w:Cmd.
(pv11\_p1\_decision\{i:l\}(Cmd;f;accpts;ldrs;ldrs$_{uid}$;reps;es;e';k\000C;w)
{}\mRightarrow{} has-es-info-type(es;e';f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num())
{}\mRightarrow{} (\muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) (snd(snd(snd(msgval(e'))))) b\000C))
{}\mRightarrow{} (w = v))))))
Date html generated:
2016_05_17-PM-04_18_02
Last ObjectModification:
2016_01_18-AM-11_17_52
Theory : paxos!synod
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