Nuprl Lemma : pv11_p1_commander_state_from_p2bs
∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀start,e,e0:E. ∀accpts,ldrs:bag(Id).
∀ldrs_uid:Id ⟶ ℤ. ∀reps:bag(Id). ∀b:pv11_p1_Ballot_Num(). ∀s:ℤ. ∀c:Cmd. ∀waitfor:bag(Id). ∀i,l:Id.
(start ≤loc e
⇒ Inj(Id;ℤ;ldrs_uid)
⇒ pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)
⇒ pv11_p1_p2a'send(Cmd;f) i <l, b, s, c> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e0)
⇒ (waitfor = pv11_p1_CommanderStateFun(Cmd;accpts;f;b;s;es.start;e) ∈ bag(Id))
⇒ (↓∃L:E List
((waitfor = [i∈accpts|¬bi ∈b map(λe.loc(e);L)] ∈ bag(Id))
∧ (∀e':E
((e' ∈ L)
⇒ ((e' < e)
∧ loc(e') ↓∈ accpts
∧ (b = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num())
∧ (<b, s, c> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')))))))))
Proof
Definitions occuring in Statement :
pv11_p1_p2a'send: pv11_p1_p2a'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_CommanderStateFun: pv11_p1_CommanderStateFun(Cmd;accpts;mf;x;zzs;es;e),
pv11_p1_AcceptorStateFun: pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;mf;es;e),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
msg-interface: Interface,
Message: Message(f),
classrel: v ∈ X(e),
eo-forward: eo.e,
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
es-causl: (e < e'),
es-loc: loc(e),
es-E: E,
id-deq: IdDeq,
Id: Id,
l_member: (x ∈ l),
deq-member: x ∈b L,
map: map(f;as),
list: T List,
inject: Inj(A;B;f),
vatype: ValueAllType,
bnot: ¬bb,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
int: ℤ,
equal: s = t ∈ T,
bag-member: x ↓∈ bs,
bag-filter: [x∈b|p[x]],
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),
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,
pv11_p1_p2b'base: pv11_p1_p2b'base(Cmd;mf),
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
encodes-msg-type: hdr encodes T,
has-es-info-type: has-es-info-type(es;e;f;T),
es-info-type: es-info-type(es;e;f),
msg-type: msg-type(msg;f),
es-header: header(e),
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
pv11_p1_on_p2b: pv11_p1_on_p2b(),
spreadn: spread4,
pi2: snd(t),
pi1: fst(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),
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(),
cand: A c∧ B,
bag-remove: bs - x,
rev_uimplies: rev_uimplies(P;Q),
deq: EqDecider(T),
es-le: e ≤loc e' ,
eq_id: a = b,
bor: p ∨bq,
label: ...$L... t
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)). \mforall{}start,e,e0:E.
\mforall{}accpts,ldrs:bag(Id). \mforall{}ldrs$_{uid}$:Id {}\mrightarrow{} \mBbbZ{}. \mforall{}reps:bag(Id). \mforall{}b:pv11\_p1\_Ballot\_Nu\000Cm(). \mforall{}s:\mBbbZ{}. \mforall{}c:Cmd.
\mforall{}waitfor:bag(Id). \mforall{}i,l:Id.
(start \mleq{}loc e
{}\mRightarrow{} Inj(Id;\mBbbZ{};ldrs$_{uid}$)
{}\mRightarrow{} pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid\mbackslash{}ff7\000Cd$; reps; f; es)
{}\mRightarrow{} pv11\_p1\_p2a'send(Cmd;f) i <l, b, s, c> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid\mbackslash{}ff\000C7d$;reps;f)(e0)
{}\mRightarrow{} (waitfor = pv11\_p1\_CommanderStateFun(Cmd;accpts;f;b;s;es.start;e))
{}\mRightarrow{} (\mdownarrow{}\mexists{}L:E List
((waitfor = [i\mmember{}accpts|\mneg{}\msubb{}i \mmember{}\msubb{} map(\mlambda{}e.loc(e);L)])
\mwedge{} (\mforall{}e':E
((e' \mmember{} L)
{}\mRightarrow{} ((e' < e)
\mwedge{} loc(e') \mdownarrow{}\mmember{} accpts
\mwedge{} (b = (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e')))\000C)
\mwedge{} (<b, s, c> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;e\000Cs;e')))))))))
Date html generated:
2016_05_17-PM-04_08_33
Last ObjectModification:
2016_01_18-AM-11_25_22
Theory : paxos!synod
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