Nuprl Lemma : pv11_p1_decision_from_p2a_acc
∀[Cmd:ValueAllType]. ∀[f:pv11_p1_headers_type{i:l}(Cmd)]. ∀[es:EO+(Message(f))]. ∀[accpts,ldrs:bag(Id)].
∀[ldrs_uid:Id ⟶ ℤ]. ∀[reps:bag(Id)]. ∀[e:E]. ∀[s:ℤ]. ∀[c:Cmd]. ∀[a:Id].
(Inj(Id;ℤ;ldrs_uid)
⇒ pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)
⇒ pv11_p1_decision{i:l}(Cmd;f;accpts;ldrs;ldrs_uid;reps;es;e;s;c)
⇒ has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
⇒ a ↓∈ accpts
⇒ (↓∃e':E
((e' < e)
∧ loc(e') ↓∈ accpts
∧ (s = (fst(snd(snd(msgval(e))))) ∈ ℤ)
∧ ((fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e))))) ∈ pv11_p1_Ballot_Num())
∧ <loc(e), snd(snd(snd(msgval(e)))), s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e'))))
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_message-constraint: pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; mf; es),
pv11_p1_p2a'base: pv11_p1_p2a'base(Cmd;mf),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
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),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-causl: (e < e'),
es-loc: loc(e),
es-E: E,
Id: Id,
inject: Inj(A;B;f),
vatype: ValueAllType,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
int: ℤ,
equal: s = t ∈ T,
bag-member: x ↓∈ bs,
bag: bag(T)
Definitions unfolded in proof :
vatype: ValueAllType,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
subtype_rel: A ⊆r B,
listp: A List+,
name: Name,
prop: ℙ,
and: P ∧ Q,
sq_stable: SqStable(P),
l_all: (∀x∈L.P[x]),
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
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: pv11_p1_decision{i:l}(Cmd;f;accpts;ldrs;ldrs_uid;reps;es;e;k;v),
exists: ∃x:A. B[x],
top: Top,
make-msg-interface: make-msg-interface(i;l;m),
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),
pv11_p1_headers_no_inputs: pv11_p1_headers_no_inputs(),
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),
cand: A c∧ B,
pv11_p1_p2a'base: pv11_p1_p2a'base(Cmd;mf),
encodes-msg-type: hdr encodes T,
uiff: uiff(P;Q),
equal-info-body: v = body(e),
pv11_p1_p2a'send: pv11_p1_p2a'send(Cmd;f),
mk-msg-interface: mk-msg-interface(l;m),
or: P ∨ Q,
assert: ↑b,
mk-msg: mk-msg(auth;hdr;val),
make-Msg: make-Msg(hdr;val)
Latex:
\mforall{}[Cmd:ValueAllType]. \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)]. \mforall{}[e:E]. \mforall{}[s:\mBbbZ{}]\000C. \mforall{}[c:Cmd]. \mforall{}[a:Id].
(Inj(Id;\mBbbZ{};ldrs$_{uid}$)
{}\mRightarrow{} pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid\mbackslash{}ff7\000Cd$; reps; f; es)
{}\mRightarrow{} pv11\_p1\_decision\{i:l\}(Cmd;f;accpts;ldrs;ldrs$_{uid}$;reps;es;e;s;c)
{}\mRightarrow{} has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num())
{}\mRightarrow{} a \mdownarrow{}\mmember{} accpts
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E
((e' < e)
\mwedge{} loc(e') \mdownarrow{}\mmember{} accpts
\mwedge{} (s = (fst(snd(snd(msgval(e))))))
\mwedge{} ((fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e))))))
\mwedge{} <loc(e), snd(snd(snd(msgval(e)))), s, c> \mmember{} pv11\_p1\_p2a'base(Cmd;f)(e'))))
Date html generated:
2016_05_17-PM-04_17_24
Last ObjectModification:
2016_01_18-AM-11_19_46
Theory : paxos!synod
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