Nuprl Lemma : new_23_sig_progress-step9-v3
∀[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))]. ∀[e:E]. ∀[n:ℤ]. ∀[c:Cmd]. ∀[faulty:bag(Id)].
(msgs-interface-delivered-with-omissions(f;es;new_23_sig_main();faulty;flrs;reps)
⇒ bag-no-repeats(Id;reps)
⇒ (#(reps) = ((coeff * flrs) + flrs + 1) ∈ ℤ)
⇒ loc(e) ↓∈ reps
⇒ (¬loc(e) ↓∈ faulty)
⇒ <n, c> ∈ new_23_sig_Proposal(Cmd;notify;propose;f)(e)
⇒ (¬↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e)))
⇒ (↓∃e':E
((((coeff * flrs) + 1)
= ||mapfilter(λe.<snd(msgval(e)), e, snd(fst(msgval(e)))>;
λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);
[e, e'])||
∈ ℤ)
∧ e ≤loc e'
∧ (↑new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0)))))
Proof
Definitions occuring in Statement :
new_23_sig_vote_with_ballot_first: new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es;e;n;r),
new_23_sig_main: new_23_sig_main(),
new_23_sig_ReplicaStateFun: new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e),
new_23_sig_Proposal: new_23_sig_Proposal(Cmd;notify;propose;f),
new_23_sig_headers_type: new_23_sig_headers_type{i:l}(Cmd;notify;propose),
set-sig-member: set-sig-member(s),
set-sig: set-sig{i:l}(Item),
msgs-interface-delivered-with-omissions: msgs-interface-delivered-with-omissions(f;es;X;faulty;failures;ids),
es-info-body: msgval(e),
Message: Message(f),
classrel: v ∈ X(e),
eo-forward: eo.e,
event-ordering+: EO+(Info),
es-interval: [e, e'],
es-le: e ≤loc e' ,
es-loc: loc(e),
es-E: E,
Id: Id,
mapfilter: mapfilter(f;P;L),
length: ||as||,
list: T List,
deq: EqDecider(T),
int_upper: {i...},
nat: ℕ,
vatype: ValueAllType,
assert: ↑b,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
lambda: λx.A[x],
pair: <a, b>,
product: x:A × B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
atom: Atom,
equal: s = t ∈ T,
bag-member: x ↓∈ bs,
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],
implies: P ⇒ Q,
member: t ∈ T,
new_23_sig_headers_type: new_23_sig_headers_type{i:l}(Cmd;notify;propose),
name: Name,
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q,
listp: A List+,
sq_stable: SqStable(P),
l_all: (∀x∈L.P[x]),
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
new_23_sig_headers: new_23_sig_headers(notify;propose),
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 ,
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),
msgs-interface-delivered-with-omissions: msgs-interface-delivered-with-omissions(f;es;X;faulty;failures;ids),
decidable: Dec(P),
has-es-info-type: has-es-info-type(es;e;f;T),
pi2: snd(t),
bag-remove-repeats: bag-remove-repeats(eq;bs),
so_apply: x[s1;s2],
irrefl: Irrefl(T;x,y.E[x; y]),
st_anti_sym: StAntiSym(T;x,y.R[x; y]),
sorted-by: sorted-by(R;L),
int_seg: {i..j-},
lelt: i ≤ j < k,
new_23_sig_vote_with_ballot_first_alt: new_23_sig_vote_with_ballot_first_alt(Cmd;notify;propose;f;es;e;e';n;r;a),
Id: Id,
mapfilter: mapfilter(f;P;L),
bag-size: #(bs),
rev_uimplies: rev_uimplies(P;Q)
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))]. \mforall{}[e:E]. \mforall{}[n:\mBbbZ{}].
\mforall{}[c:Cmd]. \mforall{}[faulty:bag(Id)].
(msgs-interface-delivered-with-omissions(f;es;new\_23\_sig\_main();faulty;flrs;reps)
{}\mRightarrow{} bag-no-repeats(Id;reps)
{}\mRightarrow{} (\#(reps) = ((coeff * flrs) + flrs + 1))
{}\mRightarrow{} loc(e) \mdownarrow{}\mmember{} reps
{}\mRightarrow{} (\mneg{}loc(e) \mdownarrow{}\mmember{} faulty)
{}\mRightarrow{} <n, c> \mmember{} new\_23\_sig\_Proposal(Cmd;notify;propose;f)(e)
{}\mRightarrow{} (\mneg{}\muparrow{}(set-sig-member(slots) n new\_23\_sig\_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e)))
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E
((((coeff * flrs) + 1)
= ||mapfilter(\mlambda{}e.<snd(msgval(e)), e, snd(fst(msgval(e)))>
\mlambda{}e'.new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e';n;0);
[e, e'])||)
\mwedge{} e \mleq{}loc e'
\mwedge{} (\muparrow{}new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e';n;0)))))
Date html generated:
2016_05_17-PM-02_39_31
Last ObjectModification:
2016_01_18-AM-09_32_15
Theory : 2!3!consensus!with!signatures
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