Nuprl Lemma : pv11_p1_ldr_fun_proposal3
∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀e:E. ∀ldrs_uid:Id ⟶ ℤ. ∀s:ℤ. ∀c:Cmd.
((<s, c> ∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e))))
⇒ (∃e':E
((e' <loc e)
∧ (∀e'':E. ((e' <loc e'') ⇒ e'' ≤loc e ⇒ (<s, c> ∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e''))))))
∧ ((<s, c> ∈ pv11_p1_propose'base(Cmd;f)(e')
∧ (¬↑(pv11_p1_in_domain(Cmd) s (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e')))))))
∨ (∃pvals:(pv11_p1_Ballot_Num() × ℤ × Cmd) List
∃b:pv11_p1_Ballot_Num()
(<fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e')), pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e')
∧ (<b, s, c> ∈ pvals)
∧ (∀b':pv11_p1_Ballot_Num(). ∀c':Cmd.
((<b', s, c'> ∈ pvals) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) b' b))))))))))
Proof
Definitions occuring in Statement :
pv11_p1_LeaderStateFun: pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e),
pv11_p1_propose'base: pv11_p1_propose'base(Cmd;mf),
pv11_p1_adopted'base: pv11_p1_adopted'base(Cmd;mf),
pv11_p1_in_domain: pv11_p1_in_domain(Cmd),
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(),
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
es-locl: (e <loc e'),
es-E: E,
Id: Id,
l_member: (x ∈ l),
list: T List,
vatype: ValueAllType,
assert: ↑b,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
int: ℤ
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,
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),
exists: ∃x:A. B[x],
cand: A c∧ B,
top: Top,
not: ¬A,
pi2: snd(t),
false: False,
pi1: fst(t)
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}f:pv11\_p1\_headers\_type\{i:l\}(Cmd). \mforall{}es:EO+(Message(f)). \mforall{}e:E. \mforall{}ldrs$_\mbackslash{}ff7\000Cbuid}$:Id {}\mrightarrow{} \mBbbZ{}.
\mforall{}s:\mBbbZ{}. \mforall{}c:Cmd.
((<s, c> \mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e))))
{}\mRightarrow{} (\mexists{}e':E
((e' <loc e)
\mwedge{} (\mforall{}e'':E
((e' <loc e'')
{}\mRightarrow{} e'' \mleq{}loc e
{}\mRightarrow{} (<s, c> \mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e'\000C'))))))
\mwedge{} ((<s, c> \mmember{} pv11\_p1\_propose'base(Cmd;f)(e')
\mwedge{} (\mneg{}\muparrow{}(pv11\_p1\_in\_domain(Cmd) s (snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid\mbackslash{}f\000Cf7d$;f;es;e')))))))
\mvee{} (\mexists{}pvals:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List
\mexists{}b:pv11\_p1\_Ballot\_Num()
(<fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e')), pvals> \mmember{}
pv11\_p1\_adopted'base(Cmd;f)(e')
\mwedge{} (<b, s, c> \mmember{} pvals)
\mwedge{} (\mforall{}b':pv11\_p1\_Ballot\_Num(). \mforall{}c':Cmd.
((<b', s, c'> \mmember{} pvals) {}\mRightarrow{} (\muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) b\000C' b))))))))))
Date html generated:
2016_05_17-PM-03_41_48
Last ObjectModification:
2015_12_29-PM-11_18_37
Theory : paxos!synod
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