Nuprl Lemma : new_23_sig_quorum_state_eq1
∀Cmd:ValueAllType. ∀notify,propose:Atom List. ∀f:new_23_sig_headers_type{i:l}(Cmd;notify;propose). ∀es:EO+(Message(f)).
∀e:E. ∀n,r:ℤ.
(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<n, r>;es;e)
= <rev(mapfilter(λe.(snd(fst(msgval(e))));
λe.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es;e;n;r);
[es-init(es;e);e)))
, rev(mapfilter(λe.(snd(msgval(e)));
λe.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es;e;n;r);
[es-init(es;e);e)))
>
∈ (Cmd List × (Id List)))
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_QuorumStateFun: new_23_sig_QuorumStateFun(Cmd;notify;propose;f;x;es;e),
new_23_sig_headers_type: new_23_sig_headers_type{i:l}(Cmd;notify;propose),
es-info-body: msgval(e),
Message: Message(f),
event-ordering+: EO+(Info),
es-init: es-init(es;e),
es-closed-open-interval: [e;e'),
es-E: E,
Id: Id,
mapfilter: mapfilter(f;P;L),
reverse: rev(as),
list: T List,
vatype: ValueAllType,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
lambda: λx.A[x],
pair: <a, b>,
product: x:A × B[x],
int: ℤ,
atom: Atom,
equal: s = t ∈ T
Definitions unfolded in proof :
vatype: ValueAllType,
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,
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,
implies: 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),
strongwellfounded: SWellFounded(R[x; y]),
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
has-es-info-type: has-es-info-type(es;e;f;T),
pi2: snd(t),
pi1: fst(t),
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_newvote: new_23_sig_newvote(Cmd),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
new_23_sig_addvote: new_23_sig_addvote(Cmd),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
new_23_sig_vote_with_ballot_first: new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es;e;n;r),
rev_uimplies: rev_uimplies(P;Q)
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{}n,r:\mBbbZ{}.
(new\_23\_sig\_QuorumStateFun(Cmd;notify;propose;f;<n, r>es;e)
= <rev(mapfilter(\mlambda{}e.(snd(fst(msgval(e))));
\mlambda{}e.new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es;e;n;r);
[es-init(es;e);e)))
, rev(mapfilter(\mlambda{}e.(snd(msgval(e)));
\mlambda{}e.new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es;e;n;r);
[es-init(es;e);e)))
>)
Date html generated:
2016_05_17-PM-02_34_53
Last ObjectModification:
2016_01_18-AM-09_36_39
Theory : 2!3!consensus!with!signatures
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