Nuprl Lemma : new_23_sig_agreement
∀[Cmd:ValueAllType]. ∀[eq:EqDecider(Cmd)]. ∀[reps,clients:bag(Id)]. ∀[coeff:{2...}]. ∀[flrs:ℕ].
∀[propose,notify:Atom List]. ∀[slots:set-sig{i:l}(ℤ)]. ∀[f:new_23_sig_headers_type{i:l}(Cmd;notify;propose)].
∀[es:EO+(Message(f))].
(eo-msg-interface-constraint(es;new_23_sig_main();new_23_sig_headers_internal();f)
⇒ bag-no-repeats(Id;reps)
⇒ (#(reps) = ((coeff * flrs) + flrs + 1) ∈ ℤ)
⇒ any v1,v2 from new_23_sig_decided'base(Cmd;notify;propose;f) satisfy
((fst(v1)) = (fst(v2)) ∈ ℤ) ⇒ ((snd(v1)) = (snd(v2)) ∈ Cmd))
Proof
Definitions occuring in Statement :
new_23_sig_headers_internal: new_23_sig_headers_internal(),
new_23_sig_main: new_23_sig_main(),
new_23_sig_decided'base: new_23_sig_decided'base(Cmd;notify;propose;f),
new_23_sig_headers_type: new_23_sig_headers_type{i:l}(Cmd;notify;propose),
set-sig: set-sig{i:l}(Item),
eo-msg-interface-constraint: eo-msg-interface-constraint(es;X;hdrs;f),
Message: Message(f),
consistent-class: consistent-class,
event-ordering+: EO+(Info),
Id: Id,
list: T List,
deq: EqDecider(T),
int_upper: {i...},
nat: ℕ,
vatype: ValueAllType,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
implies: P ⇒ Q,
product: x:A × B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
atom: Atom,
equal: s = t ∈ T,
bag-no-repeats: bag-no-repeats(T;bs),
bag-size: #(bs),
bag: bag(T)
Definitions unfolded in proof :
vatype: ValueAllType,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
new_23_sig_headers_type: new_23_sig_headers_type{i:l}(Cmd;notify;propose),
l_all: (∀x∈L.P[x]),
all: ∀x:A. B[x],
and: P ∧ Q,
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,
sq_type: SQType(T),
int_upper: {i...},
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,
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),
consistent-class: consistent-class,
pi1: fst(t),
pi2: snd(t),
decidable: Dec(P),
bag-append: as + bs,
bag-size: #(bs),
bag-no-repeats: bag-no-repeats(T;bs),
l_disjoint: l_disjoint(T;l1;l2),
mk-msg-interface: mk-msg-interface(l;m),
make-msg-interface: make-msg-interface(i;l;m),
make-Msg: make-Msg(hdr;val),
mk-msg: mk-msg(auth;hdr;val),
int_seg: {i..j-},
lelt: i ≤ j < k,
strongwellfounded: SWellFounded(R[x; y]),
new_23_sig_retry'base: new_23_sig_retry'base(Cmd;notify;propose;f),
encodes-msg-type: hdr encodes T,
has-es-info-type: has-es-info-type(es;e;f;T),
rev_uimplies: rev_uimplies(P;Q),
possible-majority: possible-majority(T;eq;L;x),
deq: EqDecider(T),
eqof: eqof(d),
l_member: (x ∈ l),
sq_or: a ↓∨ b,
eo-msg-interface-constraint: eo-msg-interface-constraint(es;X;hdrs;f),
deq-member: x ∈b L,
reduce: reduce(f;k;as),
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
strict4: strict4(F),
has-value: (a)↓,
new_23_sig_headers_internal: new_23_sig_headers_internal(),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bor: p ∨bq,
es-info-body: msgval(e),
es-info-auth: es-info-auth(es;e)
Latex:
\mforall{}[Cmd:ValueAllType]. \mforall{}[eq:EqDecider(Cmd)]. \mforall{}[reps,clients:bag(Id)]. \mforall{}[coeff:\{2...\}]. \mforall{}[flrs:\mBbbN{}].
\mforall{}[propose,notify:Atom List]. \mforall{}[slots:set-sig\{i:l\}(\mBbbZ{})].
\mforall{}[f:new\_23\_sig\_headers\_type\{i:l\}(Cmd;notify;propose)]. \mforall{}[es:EO+(Message(f))].
(eo-msg-interface-constraint(es;new\_23\_sig\_main();new\_23\_sig\_headers\_internal();f)
{}\mRightarrow{} bag-no-repeats(Id;reps)
{}\mRightarrow{} (\#(reps) = ((coeff * flrs) + flrs + 1))
{}\mRightarrow{} any v1,v2 from new\_23\_sig\_decided'base(Cmd;notify;propose;f) satisfy
((fst(v1)) = (fst(v2))) {}\mRightarrow{} ((snd(v1)) = (snd(v2))))
Date html generated:
2016_05_17-PM-02_29_40
Last ObjectModification:
2016_05_13-AM-04_10_33
Theory : 2!3!consensus!with!signatures
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