Nuprl Lemma : pv11_p1_ldr_proposal3
∀Cmd:ValueAllType. ∀f:pv11_p1_headers_type{i:l}(Cmd). ∀es:EO+(Message(f)). ∀e:E. ∀ldrs_uid:Id ⟶ ℤ.
∀v:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List). ∀s:ℤ. ∀c:Cmd.
(v ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e)
⇒ (<s, c> ∈ snd(snd(v)))
⇒ (∃e':E
∃bnum:pv11_p1_Ballot_Num()
∃active:𝔹
∃proposals:(ℤ × Cmd) List
((e' <loc e)
∧ <bnum, active, proposals> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e')
∧ (∀e'':E. ∀x:pv11_p1_Ballot_Num() × 𝔹 × ((ℤ × Cmd) List).
((e' <loc e'') ⇒ e'' ≤loc e ⇒ x ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e'') ⇒ (<s, c> ∈ snd(snd(x)))))
∧ ((<s, c> ∈ pv11_p1_propose'base(Cmd;f)(e') ∧ (¬↑(pv11_p1_in_domain(Cmd) s proposals)))
∨ (∃pvals:(pv11_p1_Ballot_Num() × ℤ × Cmd) List
∃b:pv11_p1_Ballot_Num()
(<bnum, 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_LeaderState: pv11_p1_LeaderState(Cmd;ldrs_uid;mf),
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,
bool: 𝔹,
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,
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),
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,
uiff: uiff(P;Q),
sq_type: SQType(T),
pv11_p1_init_leader: pv11_p1_init_leader(Cmd),
pv11_p1_init_proposals: pv11_p1_init_proposals(Cmd),
pi1: fst(t),
pv11_p1_on_propose: pv11_p1_on_propose(Cmd),
spreadn: spread3,
let: let,
cand: A c∧ B,
es-le: e ≤loc e' ,
es-functional-class: X is functional,
single-valued-classrel: single-valued-classrel(es;X;T),
pv11_p1_add_if_new: pv11_p1_add_if_new(),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
es-locl: (e <loc e'),
pv11_p1_when_adopted: pv11_p1_when_adopted(Cmd;ldrs_uid),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
Id: Id,
pv11_p1_when_preempted: pv11_p1_when_preempted(Cmd;ldrs_uid)
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{}v:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List). \mforall{}s:\mBbbZ{}. \mforall{}c:Cmd.
(v \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e)
{}\mRightarrow{} (<s, c> \mmember{} snd(snd(v)))
{}\mRightarrow{} (\mexists{}e':E
\mexists{}bnum:pv11\_p1\_Ballot\_Num()
\mexists{}active:\mBbbB{}
\mexists{}proposals:(\mBbbZ{} \mtimes{} Cmd) List
((e' <loc e)
\mwedge{} <bnum, active, proposals> \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(\000Ce')
\mwedge{} (\mforall{}e'':E. \mforall{}x:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbB{} \mtimes{} ((\mBbbZ{} \mtimes{} Cmd) List).
((e' <loc e'')
{}\mRightarrow{} e'' \mleq{}loc e
{}\mRightarrow{} x \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e'')
{}\mRightarrow{} (<s, c> \mmember{} snd(snd(x)))))
\mwedge{} ((<s, c> \mmember{} pv11\_p1\_propose'base(Cmd;f)(e') \mwedge{} (\mneg{}\muparrow{}(pv11\_p1\_in\_domain(Cmd) s proposals)))
\mvee{} (\mexists{}pvals:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List
\mexists{}b:pv11\_p1\_Ballot\_Num()
(<bnum, 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$_{uid}$\000C) b' b))))))))))
Date html generated:
2016_05_17-PM-03_41_39
Last ObjectModification:
2016_01_18-AM-11_26_08
Theory : paxos!synod
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