Nuprl Lemma : new_23_sig_quorum_mem
∀Cmd:ValueAllType. ∀notify,propose:Atom List. ∀f:new_23_sig_headers_type{i:l}(Cmd;notify;propose). ∀es:EO+(Message(f)).
∀e1,e2:E. ∀ni:ℤ × ℤ. ∀s1,s2:Cmd List × (Id List). ∀v:ℤ × ℤ × Cmd × Id.
((e1 <loc e2)
⇒ v ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e1)
⇒ s1 ∈ new_23_sig_QuorumState(Cmd;notify;propose;f) ni(e1)
⇒ s2 ∈ new_23_sig_QuorumState(Cmd;notify;propose;f) ni(e2)
⇒ let cmds1,locs1 = s1
in let cmds2,locs2 = s2
in let x,i = v
in let ni',c = x
in (ni = ni' ∈ (ℤ × ℤ)) ⇒ ((i ∈ locs2) ∧ ((¬(i ∈ locs1)) ⇒ (c ∈ cmds2))))
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-locl: (e <loc e'),
es-E: E,
Id: Id,
l_member: (x ∈ l),
list: T List,
vatype: ValueAllType,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
spread: spread def,
product: x:A × B[x],
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),
so_apply: x[s1;s2;s3],
new_23_sig_add_to_quorum: new_23_sig_add_to_quorum(Cmd),
new_23_sig_addvote: new_23_sig_addvote(Cmd),
pi1: fst(t),
pi2: snd(t),
new_23_sig_newvote: new_23_sig_newvote(Cmd),
decidable: Dec(P),
new_23_sig_vote'base: new_23_sig_vote'base(Cmd;notify;propose;f),
encodes-msg-type: hdr encodes T
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{}e1,e2:E. \mforall{}ni:\mBbbZ{} \mtimes{} \mBbbZ{}. \mforall{}s1,s2:Cmd List \mtimes{} (Id List). \mforall{}v:\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id.
((e1 <loc e2)
{}\mRightarrow{} v \mmember{} new\_23\_sig\_vote'base(Cmd;notify;propose;f)(e1)
{}\mRightarrow{} s1 \mmember{} new\_23\_sig\_QuorumState(Cmd;notify;propose;f) ni(e1)
{}\mRightarrow{} s2 \mmember{} new\_23\_sig\_QuorumState(Cmd;notify;propose;f) ni(e2)
{}\mRightarrow{} let cmds1,locs1 = s1
in let cmds2,locs2 = s2
in let x,i = v
in let ni',c = x
in (ni = ni') {}\mRightarrow{} ((i \mmember{} locs2) \mwedge{} ((\mneg{}(i \mmember{} locs1)) {}\mRightarrow{} (c \mmember{} cmds2))))
Date html generated:
2016_05_17-PM-02_15_19
Last ObjectModification:
2016_01_18-AM-09_34_12
Theory : 2!3!consensus!with!signatures
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