Nuprl Lemma : pv11_p1_ldr_fun_state_adopted_pred
∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀e1,e2:E. ∀ldrs_uid:Id ⟶ ℤ.
∀bnum:pv11_p1_Ballot_Num(). ∀active:𝔹. ∀proposals:(ℤ × Cmd) List. ∀pvals:(pv11_p1_Ballot_Num() × ℤ × Cmd) List.
((e1 <loc e2)
⇒ (<bnum, active, proposals>
= pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)
∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List)))
⇒ (header(e1) = ``pv11_p1 adopted`` ∈ Name)
⇒ (<bnum, pvals> = msgval(e1) ∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
⇒ (∃e:{e:E| ¬↑first(e)}
((e1 = pred(e) ∈ E)
∧ e ≤loc e2
∧ (<bnum, tt, pv11_p1_update_proposals(Cmd) proposals (pv11_p1_pmax(Cmd;ldrs_uid) pvals)>
= pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e)
∈ (pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List))))))
Proof
Definitions occuring in Statement :
pv11_p1_LeaderStateFun: pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e),
pv11_p1_update_proposals: pv11_p1_update_proposals(Cmd),
pv11_p1_pmax: pv11_p1_pmax(Cmd;ldrs_uid),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
es-info-body: msgval(e),
es-header: header(e),
Message: Message(f),
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
es-locl: (e <loc e'),
es-first: first(e),
es-pred: pred(e),
es-E: E,
Id: Id,
name: Name,
cons: [a / b],
nil: [],
list: T List,
vatype: ValueAllType,
assert: ↑b,
btrue: tt,
bool: 𝔹,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: 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,
token: "$token",
equal: s = t ∈ T
Definitions unfolded in proof :
vatype: ValueAllType,
all: ∀x:A. B[x],
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
l_all: (∀x∈L.P[x]),
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
listp: A List+,
name: Name,
pv11_p1_headers: pv11_p1_headers(),
rev_implies: P ⇐ Q,
guard: {T},
or: P ∨ Q,
uimplies: b supposing a,
pv11_p1_headers_fun: pv11_p1_headers_fun(Cmd),
name_eq: name_eq(x;y),
name-deq: NameDeq,
list-deq: list-deq(eq),
list_ind: list_ind,
cons: [a / b],
band: p ∧b q,
ifthenelse: if b then t else f fi ,
atom-deq: AtomDeq,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
nil: [],
it: ⋅,
null: null(as),
es-info-type: es-info-type(es;e;f),
msg-type: msg-type(msg;f),
es-header: header(e),
exists: ∃x:A. B[x],
cand: A c∧ B,
not: ¬A,
false: False
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)). \mforall{}e1,e2:E.
\mforall{}ldrs$_{uid}$:Id {}\mrightarrow{} \mBbbZ{}. \mforall{}bnum:pv11\_p1\_Ballot\_Num(). \mforall{}active:\mBbbB{}. \mforall{}proposals:(\mBbbZ{} \mtimes{} Cm\000Cd) List.
\mforall{}pvals:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List.
((e1 <loc e2)
{}\mRightarrow{} (<bnum, active, proposals> = pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e\000C1))
{}\mRightarrow{} (header(e1) = ``pv11\_p1 adopted``)
{}\mRightarrow{} (<bnum, pvals> = msgval(e1))
{}\mRightarrow{} (\mexists{}e:\{e:E| \mneg{}\muparrow{}first(e)\}
((e1 = pred(e))
\mwedge{} e \mleq{}loc e2
\mwedge{} (<bnum, tt, pv11\_p1\_update\_proposals(Cmd) proposals (pv11\_p1\_pmax(Cmd;ldrs$_{ui\000Cd}$) pvals)>
= pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e)))))
Date html generated:
2016_05_17-PM-03_44_56
Last ObjectModification:
2015_12_29-PM-11_16_52
Theory : paxos!synod
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