Nuprl Lemma : new_23_sig-notify
∀[Cmd:{T:Type| valueall-type(T)} ]. ∀[clients:bag(Id)]. ∀[cmdeq:EqDecider(Cmd)]. ∀[coeff,flrs:ℤ].
∀[notify,propose:Atom List]. ∀[reps:bag(Id)]. ∀[slots:set-sig{i:l}(ℤ)].
∀[f:new_23_sig_headers_type{i:l}(Cmd;notify;propose)]. ∀[es:EO+(Message(f))]. ∀[e:E]. ∀[d:ℤ]. ∀[i:Id]. ∀[auth:𝔹].
∀[k:ℤ]. ∀[v:Cmd].
(<d, i, mk-msg(auth;notify;<k, v>)> ∈ new_23_sig_main()(e)
⇐⇒ loc(e) ↓∈ reps
∧ (↓∃z2:Cmd
∃e':{e':E| e' ≤loc e }
((header(e) = ``new_23_sig decided`` ∈ Name)
∧ has-es-info-type(es;e;f;ℤ × Cmd)
∧ i ↓∈ clients
∧ (d = 0 ∈ ℤ)
∧ auth = ff
∧ (<k, v> = msgval(e) ∈ (f notify))
∧ (((header(e') = propose ∈ Name)
∧ has-es-info-type(es;e';f;ℤ × Cmd)
∧ (<fst(msgval(e)), z2> = msgval(e') ∈ (ℤ × Cmd)))
∨ ((header(e') = ``new_23_sig vote`` ∈ Name)
∧ has-es-info-type(es;e';f;ℤ × ℤ × Cmd × Id)
∧ ((fst(msgval(e))) = (fst(fst(fst(msgval(e'))))) ∈ ℤ)
∧ (z2 = (snd(fst(msgval(e')))) ∈ Cmd)))
∧ (¬↑(set-sig-member(slots) (fst(msgval(e))) new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e')))
∧ (no (new_23_sig_decision(Cmd;clients;notify;propose;f)
(fst(msgval(e))) o new_23_sig_decided'base(Cmd;notify;propose;f)) between e' and e))))
Proof
Definitions occuring in Statement :
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_decision: new_23_sig_decision(Cmd;clients;notify;propose;f),
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-member: set-sig-member(s),
set-sig: set-sig{i:l}(Item),
msg-interface: Interface,
mk-msg: mk-msg(auth;hdr;val),
es-info-body: msgval(e),
has-es-info-type: has-es-info-type(es;e;f;T),
es-header: header(e),
Message: Message(f),
eclass0: (f o X),
no-classrel-in-interval: (no X between start and e),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
es-loc: loc(e),
es-E: E,
Id: Id,
name: Name,
cons: [a / b],
nil: [],
list: T List,
deq: EqDecider(T),
valueall-type: valueall-type(T),
assert: ↑b,
bfalse: ff,
bool: 𝔹,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
squash: ↓T,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
pair: <a, b>,
product: x:A × B[x],
natural_number: $n,
int: ℤ,
token: "$token",
atom: Atom,
universe: Type,
equal: s = t ∈ T,
bag-member: x ↓∈ bs,
bag: bag(T)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
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,
vatype: ValueAllType,
prop: ℙ,
and: P ∧ Q,
listp: A List+,
implies: P ⇒ Q,
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,
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),
bag-member: x ↓∈ bs,
make-msg-interface: make-msg-interface(i;l;m),
classrel: v ∈ X(e),
has-es-info-type: has-es-info-type(es;e;f;T),
pi1: fst(t),
pi2: snd(t),
equal-info-body: v = body(e)
Latex:
\mforall{}[Cmd:\{T:Type| valueall-type(T)\} ]. \mforall{}[clients:bag(Id)]. \mforall{}[cmdeq:EqDecider(Cmd)]. \mforall{}[coeff,flrs:\mBbbZ{}].
\mforall{}[notify,propose:Atom List]. \mforall{}[reps:bag(Id)]. \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{}[d:\mBbbZ{}].
\mforall{}[i:Id]. \mforall{}[auth:\mBbbB{}]. \mforall{}[k:\mBbbZ{}]. \mforall{}[v:Cmd].
(<d, i, mk-msg(auth;notify;<k, v>)> \mmember{} new\_23\_sig\_main()(e)
\mLeftarrow{}{}\mRightarrow{} loc(e) \mdownarrow{}\mmember{} reps
\mwedge{} (\mdownarrow{}\mexists{}z2:Cmd
\mexists{}e':\{e':E| e' \mleq{}loc e \}
((header(e) = ``new\_23\_sig decided``)
\mwedge{} has-es-info-type(es;e;f;\mBbbZ{} \mtimes{} Cmd)
\mwedge{} i \mdownarrow{}\mmember{} clients
\mwedge{} (d = 0)
\mwedge{} auth = ff
\mwedge{} (<k, v> = msgval(e))
\mwedge{} (((header(e') = propose)
\mwedge{} has-es-info-type(es;e';f;\mBbbZ{} \mtimes{} Cmd)
\mwedge{} (<fst(msgval(e)), z2> = msgval(e')))
\mvee{} ((header(e') = ``new\_23\_sig vote``)
\mwedge{} has-es-info-type(es;e';f;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)
\mwedge{} ((fst(msgval(e))) = (fst(fst(fst(msgval(e'))))))
\mwedge{} (z2 = (snd(fst(msgval(e')))))))
\mwedge{} (\mneg{}\muparrow{}(set-sig-member(slots) (fst(msgval(e)))
new\_23\_sig\_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e')))
\mwedge{} (no (new\_23\_sig\_decision(Cmd;clients;notify;propose;f)
(fst(msgval(e))) o new\_23\_sig\_decided'base(Cmd;notify;propose;f)) between e' and
e))))
Date html generated:
2016_05_17-PM-02_14_28
Last ObjectModification:
2016_01_18-PM-07_20_53
Theory : 2!3!consensus!with!signatures
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