Nuprl Lemma : new_23_sig_retry_property
∀[Cmd:ValueAllType]. ∀[eq:EqDecider(Cmd)]. ∀[reps,clients:bag(Id)]. ∀[coeff,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)
⇒ (∀e:E. ∀c:Cmd. ∀n,r:ℤ.
(<<n, r + 1>, c> ∈ new_23_sig_retry'base(Cmd;notify;propose;f)(e)
⇒ (↓∃x:Id
∃cs:Cmd List+
∃L:Id List
((no_repeats(Id;L) ∧ (||L|| = ((coeff * flrs) + 1) ∈ ℤ) ∧ (||cs|| = ||L|| ∈ ℤ))
∧ (∀i:ℕ||L||. ((↓∃e':E. mk-msg-interface(x;make-Msg(``new_23_sig vote``;<<<n, r>, cs[i]>, L[i]>)) ∈ new_\000C23_sig_main()(e')) ∧ L[i] ↓∈ reps))
∧ (c = (snd(poss-maj(eq;cs;hd(cs)))) ∈ 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_retry'base: new_23_sig_retry'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),
mk-msg-interface: mk-msg-interface(l;m),
msg-interface: Interface,
make-Msg: make-Msg(hdr;val),
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-E: E,
Id: Id,
poss-maj: poss-maj(eq;L;x),
no_repeats: no_repeats(T;l),
select: L[n],
listp: A List+,
hd: hd(l),
length: ||as||,
cons: [a / b],
nil: [],
list: T List,
deq: EqDecider(T),
int_seg: {i..j-},
nat: ℕ,
vatype: ValueAllType,
uall: ∀[x:A]. B[x],
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
pair: <a, b>,
product: x:A × B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
token: "$token",
atom: Atom,
equal: s = t ∈ T,
bag-member: x ↓∈ bs,
bag: bag(T)
Definitions unfolded in proof :
vatype: ValueAllType,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
new_23_sig_headers_type: new_23_sig_headers_type{i:l}(Cmd;notify;propose),
l_all: (∀x∈L.P[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,
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),
eo-msg-interface-constraint: eo-msg-interface-constraint(es;X;hdrs;f),
new_23_sig_retry'base: new_23_sig_retry'base(Cmd;notify;propose;f),
es-header: header(e),
new_23_sig_headers_internal: new_23_sig_headers_internal(),
make-msg-interface: make-msg-interface(i;l;m),
has-es-info-type: has-es-info-type(es;e;f;T),
pi2: snd(t),
pi1: fst(t),
decidable: Dec(P),
int_seg: {i..j-},
lelt: i ≤ j < k,
es-info-body: msgval(e),
es-info-auth: es-info-auth(es;e),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
mk-msg: mk-msg(auth;hdr;val),
make-Msg: make-Msg(hdr;val),
mk-msg-interface: mk-msg-interface(l;m),
new_23_sig_vote'base: new_23_sig_vote'base(Cmd;notify;propose;f),
encodes-msg-type: hdr encodes T,
Id: Id,
es-locl: (e <loc e'),
bag-member: x ↓∈ bs
Latex:
\mforall{}[Cmd:ValueAllType]. \mforall{}[eq:EqDecider(Cmd)]. \mforall{}[reps,clients:bag(Id)]. \mforall{}[coeff,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{} (\mforall{}e:E. \mforall{}c:Cmd. \mforall{}n,r:\mBbbZ{}.
(<<n, r + 1>, c> \mmember{} new\_23\_sig\_retry'base(Cmd;notify;propose;f)(e)
{}\mRightarrow{} (\mdownarrow{}\mexists{}x:Id
\mexists{}cs:Cmd List\msupplus{}
\mexists{}L:Id List
((no\_repeats(Id;L) \mwedge{} (||L|| = ((coeff * flrs) + 1)) \mwedge{} (||cs|| = ||L||))
\mwedge{} (\mforall{}i:\mBbbN{}||L||
((\mdownarrow{}\mexists{}e':E
mk-msg-interface(x;make-Msg(``new\_23\_sig vote``;<<<n, r>, cs[i]>, L[i]>)) \mmember{}\000C new\_23\_sig\_main()(e'))
\mwedge{} L[i] \mdownarrow{}\mmember{} reps))
\mwedge{} (c = (snd(poss-maj(eq;cs;hd(cs))))))))))
Date html generated:
2016_05_17-PM-02_27_45
Last ObjectModification:
2016_05_13-AM-03_15_26
Theory : 2!3!consensus!with!signatures
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