Nuprl Lemma : new_23_sig_quorum_inv_vote2
∀Cmd:ValueAllType. ∀notify,propose:Atom List. ∀f:new_23_sig_headers_type{i:l}(Cmd;notify;propose). ∀es:EO+(Message(f)).
∀e:E. ∀ni:ℤ × ℤ. ∀p:Cmd List × (Id List).
(p ∈ new_23_sig_QuorumState(Cmd;notify;propose;f) ni(e)
⇒ let cmds,locs = p
in no_repeats(Id;locs)
∧ (||locs|| = ||cmds|| ∈ ℤ)
∧ (∀i:ℕ||locs||
(↓∃e':E
((e' <loc e)
∧ <<ni, cmds[i]>, locs[i]> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e')
∧ (∀e''∈[es-init(es;e);e').∀c:Cmd
(¬<<ni, c>, locs[i]> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(
e'')))))))
Proof
Definitions occuring in Statement :
new_23_sig_QuorumState: new_23_sig_QuorumState(Cmd;notify;propose;f),
new_23_sig_vote'base: new_23_sig_vote'base(Cmd;notify;propose;f),
new_23_sig_headers_type: new_23_sig_headers_type{i:l}(Cmd;notify;propose),
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-init: es-init(es;e),
es-closed-open-interval: [e;e'),
es-locl: (e <loc e'),
es-E: E,
Id: Id,
l_all: (∀x∈L.P[x]),
no_repeats: no_repeats(T;l),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
vatype: ValueAllType,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
spread: spread def,
pair: <a, b>,
product: x:A × B[x],
natural_number: $n,
int: ℤ,
atom: Atom,
equal: s = t ∈ T
Definitions unfolded in proof :
vatype: ValueAllType,
all: ∀x:A. B[x],
implies: P ⇒ Q,
new_23_sig_headers_type: new_23_sig_headers_type{i:l}(Cmd;notify;propose),
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,
name: Name,
new_23_sig_headers: new_23_sig_headers(notify;propose),
listp: A List+,
rev_implies: P ⇐ Q,
guard: {T},
or: P ∨ Q,
uimplies: b supposing a,
new_23_sig_headers_fun: new_23_sig_headers_fun(Cmd;notify;propose),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
sq_stable: SqStable(P),
cand: A c∧ B,
new_23_sig_headers_no_rep: new_23_sig_headers_no_rep(notify;propose),
squash: ↓T,
no_repeats: no_repeats(T;l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
length: ||as||,
list_ind: list_ind,
cons: [a / b],
nil: [],
less_than: a < b,
true: True,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
select: L[n],
subtract: n - m,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
name_eq: name_eq(x;y),
name-deq: NameDeq,
list-deq: list-deq(eq),
band: p ∧b q,
atom-deq: AtomDeq,
eq_atom: x =a y,
null: null(as),
new_23_sig_QuorumState: new_23_sig_QuorumState(Cmd;notify;propose;f),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
Memory1: Memory1(init;tr;X),
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
new_23_sig_vote'base: new_23_sig_vote'base(Cmd;notify;propose;f),
encodes-msg-type: hdr encodes T,
new_23_sig_add_to_quorum: new_23_sig_add_to_quorum(Cmd),
new_23_sig_addvote: new_23_sig_addvote(Cmd),
rev_uimplies: rev_uimplies(P;Q),
new_23_sig_newvote: new_23_sig_newvote(Cmd)
Latex:
\mforall{}Cmd:ValueAllType. \mforall{}notify,propose:Atom List. \mforall{}f:new\_23\_sig\_headers\_type\{i:l\}(Cmd;notify;propose).
\mforall{}es:EO+(Message(f)). \mforall{}e:E. \mforall{}ni:\mBbbZ{} \mtimes{} \mBbbZ{}. \mforall{}p:Cmd List \mtimes{} (Id List).
(p \mmember{} new\_23\_sig\_QuorumState(Cmd;notify;propose;f) ni(e)
{}\mRightarrow{} let cmds,locs = p
in no\_repeats(Id;locs)
\mwedge{} (||locs|| = ||cmds||)
\mwedge{} (\mforall{}i:\mBbbN{}||locs||
(\mdownarrow{}\mexists{}e':E
((e' <loc e)
\mwedge{} <<ni, cmds[i]>, locs[i]> \mmember{} new\_23\_sig\_vote'base(Cmd;notify;propose;f)(e')
\mwedge{} (\mforall{}e''\mmember{}[es-init(es;e);e').\mforall{}c:Cmd
(\mneg{}<<ni, c>, locs[i]> \mmember{}
new\_23\_sig\_vote'base(Cmd;notify;propose;f)(
e'')))))))
Date html generated:
2016_05_17-PM-02_15_28
Last ObjectModification:
2016_01_18-AM-09_35_59
Theory : 2!3!consensus!with!signatures
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