Nuprl Lemma : pv11_p1-adopted
∀[Cmd:{T:Type| valueall-type(T)} ]. ∀[accpts,ldrs:bag(Id)]. ∀[ldrs_uid:Id ⟶ ℤ]. ∀[reps:bag(Id)].
∀[mf:pv11_p1_headers_type{i:l}(Cmd)]. ∀[es:EO+(Message(mf))]. ∀[e:E]. ∀[d:ℤ]. ∀[i:Id]. ∀[auth:𝔹].
∀[p:pv11_p1_Ballot_Num()]. ∀[L:(pv11_p1_Ballot_Num() × ℤ × Cmd) List].
(<d, i, mk-msg(auth;``pv11_p1 adopted``;<p, L>)> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;mf)(e)
⇐⇒ loc(e) ↓∈ ldrs
∧ (↓((no ((pv11_p1_scout_output(Cmd;accpts;mf) (pv11_p1_init_ballot_num() loc(e)) o pv11_p1_p1b'base(Cmd;mf)) o
pv11_p1_ScoutState(Cmd;accpts;mf) (pv11_p1_init_ballot_num() loc(e))) prior to e)
∧ ((header(e) = ``pv11_p1 p1b`` ∈ Name)
∧ has-es-info-type(es;e;mf;Id
× pv11_p1_Ballot_Num()
× pv11_p1_Ballot_Num()
× ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
∧ ((pv11_p1_init_ballot_num() loc(e)) = (fst(snd(msgval(e)))) ∈ pv11_p1_Ballot_Num())
∧ (d = 0 ∈ ℤ)
∧ (i = loc(e) ∈ Id)
∧ ((pv11_p1_init_ballot_num() loc(e)) = (fst(snd(snd(msgval(e))))) ∈ pv11_p1_Ballot_Num())
∧ #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;mf;pv11_p1_init_ballot_num()
loc(e);es;e))) < pv11_p1_threshold(accpts)
∧ auth = pv11_p1_init_active()
∧ (p = (pv11_p1_init_ballot_num() loc(e)) ∈ pv11_p1_Ballot_Num())
∧ (L
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;mf;pv11_p1_init_ballot_num() loc(e);es;e)))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
∨ (∃e':{e':E| e' ≤loc e }
((((header(e') = ``pv11_p1 preempted`` ∈ Name) ∧ has-es-info-type(es;e';mf;pv11_p1_Ballot_Num()))
∧ (↑(pv11_p1_is_bnum() msgval(e')))
∧ (↑(fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e')) < msgval(e'))))
∧ (no ((pv11_p1_scout_output(Cmd;accpts;mf) (pv11_p1_upd_bnum() msgval(e') loc(e')) o
pv11_p1_p1b'base(Cmd;mf)) o pv11_p1_ScoutState(Cmd;accpts;mf)
(pv11_p1_upd_bnum() msgval(e') loc(e'))) between e' and e)
∧ ((header(e) = ``pv11_p1 p1b`` ∈ Name)
∧ has-es-info-type(es;e;mf;Id
× pv11_p1_Ballot_Num()
× pv11_p1_Ballot_Num()
× ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
∧ ((pv11_p1_upd_bnum() msgval(e') loc(e')) = (fst(snd(msgval(e)))) ∈ pv11_p1_Ballot_Num())
∧ (d = 0 ∈ ℤ)
∧ (i = loc(e) ∈ Id)
∧ ((pv11_p1_upd_bnum() msgval(e') loc(e')) = (fst(snd(snd(msgval(e))))) ∈ pv11_p1_Ballot_Num())
∧ #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;mf;pv11_p1_upd_bnum() msgval(e')
loc(e');es.e';e))) < pv11_p1_threshold(accpts)
∧ auth = pv11_p1_init_active()
∧ (p = (pv11_p1_upd_bnum() msgval(e') loc(e')) ∈ pv11_p1_Ballot_Num())
∧ (L
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;mf;pv11_p1_upd_bnum() msgval(e') loc(e');es.e';e)))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))))))
Proof
Definitions occuring in Statement :
pv11_p1_main: pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;mf),
pv11_p1_LeaderStateFun: pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e),
pv11_p1_scout_output: pv11_p1_scout_output(Cmd;accpts;mf),
pv11_p1_ScoutStateFun: pv11_p1_ScoutStateFun(Cmd;accpts;mf;x;es;e),
pv11_p1_ScoutState: pv11_p1_ScoutState(Cmd;accpts;mf),
pv11_p1_init_active: pv11_p1_init_active(),
pv11_p1_init_ballot_num: pv11_p1_init_ballot_num(),
pv11_p1_p1b'base: pv11_p1_p1b'base(Cmd;mf),
pv11_p1_threshold: pv11_p1_threshold(accpts),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_lt_bnum: pv11_p1_lt_bnum(ldrs_uid),
pv11_p1_upd_bnum: pv11_p1_upd_bnum(),
pv11_p1_is_bnum: pv11_p1_is_bnum(),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
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),
eclass2: (X o Y),
eclass1: (f o X),
no-classrel-in-interval: (no X between start and e),
no-prior-classrel: (no X prior to e),
classrel: v ∈ X(e),
eo-forward: eo.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,
valueall-type: valueall-type(T),
assert: ↑b,
bool: 𝔹,
less_than: a < b,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
squash: ↓T,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
natural_number: $n,
int: ℤ,
token: "$token",
universe: Type,
equal: s = t ∈ T,
bag-member: x ↓∈ bs,
bag-size: #(bs),
bag: bag(T)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
subtype_rel: A ⊆r B,
listp: A List+,
name: Name,
vatype: ValueAllType,
prop: ℙ,
and: P ∧ Q,
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,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
length: ||as||,
list_ind: list_ind,
pv11_p1_headers: pv11_p1_headers(),
cons: [a / b],
nil: [],
it: ⋅,
true: True,
select: L[n],
subtract: n - m,
uimplies: b supposing a,
guard: {T},
rev_implies: P ⇐ Q,
pv11_p1_headers_fun: pv11_p1_headers_fun(Cmd),
name_eq: name_eq(x;y),
name-deq: NameDeq,
list-deq: list-deq(eq),
band: p ∧b q,
ifthenelse: if b then t else f fi ,
atom-deq: AtomDeq,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
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),
top: Top,
nat: ℕ,
exists: ∃x:A. B[x],
or: P ∨ Q,
cand: A c∧ B,
pv11_p1_headers_no_rep: pv11_p1_headers_no_rep(),
no_repeats: no_repeats(T;l),
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
hint: hint(t),
pv11_p1_init_active: pv11_p1_init_active()
Latex:
\mforall{}[Cmd:\{T:Type| valueall-type(T)\} ]. \mforall{}[accpts,ldrs:bag(Id)]. \mforall{}[ldrs$_{uid}$:Id {}\mrightarrow{}\000C \mBbbZ{}]. \mforall{}[reps:bag(Id)].
\mforall{}[mf:pv11\_p1\_headers\_type\{i:l\}(Cmd)]. \mforall{}[es:EO+(Message(mf))]. \mforall{}[e:E]. \mforall{}[d:\mBbbZ{}]. \mforall{}[i:Id]. \mforall{}[auth:\mBbbB{}].
\mforall{}[p:pv11\_p1\_Ballot\_Num()]. \mforall{}[L:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List].
(<d, i, mk-msg(auth;``pv11\_p1 adopted``;<p, L>)> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_\mbackslash{}ff7\000Cbuid}$;reps;mf)(e)
\mLeftarrow{}{}\mRightarrow{} loc(e) \mdownarrow{}\mmember{} ldrs
\mwedge{} (\mdownarrow{}((no ((pv11\_p1\_scout\_output(Cmd;accpts;mf) (pv11\_p1\_init\_ballot\_num() loc(e)) o
pv11\_p1\_p1b'base(Cmd;mf)) o pv11\_p1\_ScoutState(Cmd;accpts;mf)
(pv11\_p1\_init\_ballot\_num() loc(e))) prior to e)
\mwedge{} ((header(e) = ``pv11\_p1 p1b``)
\mwedge{} has-es-info-type(es;e;mf;Id
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)))
\mwedge{} ((pv11\_p1\_init\_ballot\_num() loc(e)) = (fst(snd(msgval(e)))))
\mwedge{} (d = 0)
\mwedge{} (i = loc(e))
\mwedge{} ((pv11\_p1\_init\_ballot\_num() loc(e)) = (fst(snd(snd(msgval(e))))))
\mwedge{} \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;mf;pv11\_p1\_init\_ballot\_num()
loc(e);es;e))) < pv11\_p1\_threshold(accpts)
\mwedge{} auth = pv11\_p1\_init\_active()
\mwedge{} (p = (pv11\_p1\_init\_ballot\_num() loc(e)))
\mwedge{} (L = (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;mf;pv11\_p1\_init\_ballot\_num() loc(e);es;e)))))
\mvee{} (\mexists{}e':\{e':E| e' \mleq{}loc e \}
((((header(e') = ``pv11\_p1 preempted``)
\mwedge{} has-es-info-type(es;e';mf;pv11\_p1\_Ballot\_Num()))
\mwedge{} (\muparrow{}(pv11\_p1\_is\_bnum() msgval(e')))
\mwedge{} (\muparrow{}(fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;mf;es;e')) < ms\000Cgval(e'))))
\mwedge{} (no ((pv11\_p1\_scout\_output(Cmd;accpts;mf) (pv11\_p1\_upd\_bnum() msgval(e') loc(e')) o
pv11\_p1\_p1b'base(Cmd;mf)) o pv11\_p1\_ScoutState(Cmd;accpts;mf)
(pv11\_p1\_upd\_bnum() msgval(e') loc(e'))) between e'
and e)
\mwedge{} ((header(e) = ``pv11\_p1 p1b``)
\mwedge{} has-es-info-type(es;e;mf;Id
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)))
\mwedge{} ((pv11\_p1\_upd\_bnum() msgval(e') loc(e')) = (fst(snd(msgval(e)))))
\mwedge{} (d = 0)
\mwedge{} (i = loc(e))
\mwedge{} ((pv11\_p1\_upd\_bnum() msgval(e') loc(e')) = (fst(snd(snd(msgval(e))))))
\mwedge{} \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;mf;pv11\_p1\_upd\_bnum() msgval(e')
loc(e');es.e';e))) < ...
\mwedge{} auth = pv11\_p1\_init\_active()
\mwedge{} (p = (pv11\_p1\_upd\_bnum() msgval(e') loc(e')))
\mwedge{} (L
= (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;mf;pv11\_p1\_upd\_bnum() msgval(e')
loc(e');es.e';e))))))))
Date html generated:
2016_05_17-PM-03_11_41
Last ObjectModification:
2016_01_18-PM-07_01_54
Theory : paxos!synod
Home
Index