Nuprl Lemma : Paxos-spec8-spec7
[Info:Type]
es:EO+(Info)
[T:Type]
leader:
Id. failset:Id List. 1a:EClass(Id 
). X,Y:EClass(T).
(Paxos-spec8{i:l}(Info; es; T; leader; failset; 1a; X; Y)
Paxos-spec7{i:l}(Info; es; T; leader; failset; (snd(p) where p from 1a); X; Y))
Proof not projected
Definitions occuring in Statement :
Paxos-spec8: Paxos-spec8{i:l}(Info; es; T; leader; failset; 1a; Decide; Input),
Paxos-spec7: Paxos-spec7{i:l}(Info; es; T; leader; failset; Reserve; Decide; Input),
map-class: (f[v] where v from X),
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
Id: Id,
nat: ,
uall: [x:A]. B[x],
pi2: snd(t),
all: x:A. B[x],
implies: P Q,
function: x:A B[x],
product: x:A B[x],
list: type List,
universe: Type
Definitions :
btrue: tt,
es-collect-filter-max: Collect(size v's from X with maximum num[v] such that P[v] return <num[v],n,v> with n = maximum f[v]),
paxos-state-value: Value(s),
paxos-state-ballot: Ballot(s),
spreadn: spread4,
paxos-state-name: Name(s),
paxos-state-reservation: Reservation(s),
MaxVote: MaxVote(es;T;Vote;e;s),
es-class-causal-rel-fail: es-class-causal-rel-fail,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
es-class-causal-mrel-fail: es-class-causal-mrel-fail,
l_all: (x
L.P[x]),
es-causle: e c
e',
decision: Decision,
accum_list: accum_list(a,x.f[a; x];x.base[x];L),
max-f-class: (v from X with maximum f[v]),
so_apply: x[s],
list_accum: list_accum(x,a.f[x; a];y;l),
rev_implies: P 
Q,
iff: P Q,
p-outcome: Outcome,
less: if (a) < (b) then c else d,
base: Base,
grp_car: |g|,
es-prior-val: (X)',
es-interface-predecessors: (X)(e),
outl: outl(x),
isl: isl(x),
list-max: list-max(x.f[x];L),
int_nzero: 
,
es-first-at: e is first@ i s.t. e.P[e],
id-deq: IdDeq,
deq-member: deq-member(eq;x;L),
filter: filter(P;l),
es-locl: (e <loc e'),
alle-lt: e<e'.P[e],
bfalse: ff,
es-le: e loc e' ,
or: P 
Q,
es-collect-filter: Collect(size v's from X with maximum num[v] such that P[v]),
es-interface-pair-prior: X;Y,
eq_int: (i =
j),
mapfilter-class: (f[v] where v from X such that P[v]),
es-class-mcast-fail: es-class-mcast-fail,
es-class-reply-or-fail: es-class-reply-or-fail,
accum-class: accum-class(a,x.f[a; x];x.base[x];X),
es-tagged-true-class: Tagged_tt(X),
minus: -n,
es-prior-class-when: (X'?d) when Y,
imax: imax(a;b),
es-interface-accum: es-interface-accum(f;x;X),
le_int: i z j,
it: 
,
inr: inr x ,
inl: inl x ,
lt_int: i <z j,
es-filter-image: f[X],
in-eclass: e 
X,
eclass-val: X(e),
es-loc: loc(e),
multiply: n * m,
length: ||as||,
add: n + m,
es-collect-opt-max: es-collect-opt-max,
decide: case b of inl(x) => s[x] | inr(y) => t[y],
sqequal: s ~ t,
natural_number: $n,
real: 
,
rationals: 
,
atom: Atom,
top: Top,
es-base-E: es-base-E(es),
token: "$token",
ifthenelse: if b then t else f fi ,
record-select: r.x,
es-E-interface: E(X),
guard: {T},
sq_type: SQType(T),
l_member: (x l),
spread: spread def,
fpf: a:A fp-> B[a],
void: Void,
atom: Atom$n,
false: False,
apply: f a,
let: let,
no_repeats: no_repeats(T;l),
assert: 
b,
Paxos-spec8-body: Paxos-spec8-body{i:l}(es; Info; T; f; acceptors; leader; Input; Decide; 1a; 1b; 2a; 2b; failset),
subtype: S 
T,
event_ordering: EO,
es-E: E,
lambda: 
x.A[x],
equal: s = t,
member: t T,
strong-subtype: strong-subtype(A;B),
eq_atom: x =a y,
eq_atom: eq_atom$n(x;y),
dep-isect: Error :dep-isect,
record+: record+,
le: A B,
ge: i 
j ,
not: 
A,
less_than: a < b,
uimplies: b supposing a,
and: P 
Q,
uiff: uiff(P;Q),
subtype_rel: A r B,
Paxos-spec7-body: Paxos-spec7-body,
so_lambda: 
x.t[x],
prop: 
,
Paxos-spec8: Paxos-spec8{i:l}(Info; es; T; leader; failset; 1a; Decide; Input),
event-ordering+: EO+(Info),
uall: [x:A]. B[x],
isect: 
x:A. B[x],
implies: P Q,
Paxos-spec7: Paxos-spec7{i:l}(Info; es; T; leader; failset; Reserve; Decide; Input),
exists: 
x:A. B[x],
nat_plus: 
,
set: {x:A| B[x]} ,
universe: Type,
list: type List,
so_lambda: x y.t[x; y],
all: x:A. B[x],
function: x:A B[x],
Try: Error :Try,
tactic: Error :tactic,
RepeatFor: Error :RepeatFor,
ParallelOp: Error :ParallelOp,
spreadn: spread3,
paxos-acceptor-state: AcceptorState,
abbreviation: VoteState,
Auto: Error :Auto,
D: Error :D,
int: ,
abbreviation: Collect1,
abbreviation: NoProposal,
abbreviation: Proposal,
abbreviation: MaxReserve,
abbreviation: AcceptOrReject,
MaAuto: Error :MaAuto,
abbreviation: AcceptOrRejectTaggedTrue,
abbreviation: Accept,
abbreviation: MaxAccept,
union: left + right,
unit: Unit,
abbreviation: Vote,
ExRepD: Error :ExRepD,
CollapseTHEN: Error :CollapseTHEN,
eclass: EClass(A[eo; e]),
product: x:A B[x],
Id: Id,
nat: ,
bool: 
,
abbreviation: Vote',
CollapseTHENA: Error :CollapseTHENA,
map-class: (f[v] where v from X),
pair: <a, b>,
pi2: snd(t),
pi1: fst(t),
sum-map: 
f[x] for x L,
sum: (f[x] | x < k),
true: True,
rev_uimplies: rev_uimplies(P;Q),
cand: A c B,
squash: 
T,
p-compose: f o g,
paxos-state-info: Info(s),
infix_ap: x f y,
es-causl: (e < e'),
can-apply: can-apply(f;x),
fpf-cap: f(x)?z,
limited-type: LimitedType,
decide_bfalse: decide_bfalse{decide_bfalse_compseq_tag_def:o}(v11.g[v11]; v21.f[v21]),
eq_bool: p =b q,
null: null(as),
set_blt: a < b,
grp_blt: a < b,
dcdr-to-bool: [d],
bl-all: (xL.P[x])_b,
bl-exists: (xL.P[x])_b,
b-exists: (i<n.P[i])_b,
eq_type: eq_type(T;T'),
qeq: qeq(r;s),
q_less: q_less(r;s),
q_le: q_le(r;s),
deq-disjoint: deq-disjoint(eq;as;bs),
deq-all-disjoint: deq-all-disjoint(eq;ass;bs),
eq_str: Error :eq_str,
eq_id: a = b,
eq_lnk: a = b,
es-eq-E: e = e',
es-bless: e <loc e',
es-ble: e loc e',
bimplies: p q,
band: p q,
bor: p q,
max-fst-class: MaxFst(X),
bnot: b,
map: map(f;as),
imax-list: imax-list(L),
es-init: es-init(es;e),
es-pred: pred(e),
existse-before: e<e'.P[e],
existse-le: ee'.P[e],
alle-le: ee'.P[e],
alle-between1: e[e1,e2).P[e],
existse-between1: e[e1,e2).P[e],
alle-between2: e[e1,e2].P[e],
existse-between2: e[e1,e2].P[e],
existse-between3: e(e1,e2].P[e],
es-fset-loc: i locs(s),
es-r-immediate-pred: es-r-immediate-pred(es;R;e';e),
same-thread: same-thread(es;p;e;e'),
collect-event: collect-event(es;X;n;v.num[v];L.P[L];e),
cut-order: a (X;f) b,
path-goes-thru: x-f*-y thru i,
ses-legal-sequence: Legal(pas) given prvt,
ses-action: Action(e),
decidable: Dec(P),
lg-edge: lg-edge(g;a;b),
record: record(x.T[x]),
is_list_splitting: is_list_splitting(T;L;LL;L2;f),
is_accum_splitting: is_accum_splitting(T;A;L;LL;L2;f;g;x),
req: x = y,
rnonneg: rnonneg(r),
rleq: x y,
i-member: r I,
partitions: partitions(I;p),
modulus-of-ccontinuity: modulus-of-ccontinuity(omega;I;f),
fpf-sub: f g,
sq_stable: SqStable(P),
imax-class: (maximum f[v] lb with v from X),
deq: EqDecider(T),
ma-state: State(ds),
fpf-dom: x dom(f),
es-interface-at: X@i,
es-interface-part: (X|g=i),
class-program: ClassProgram(T),
es-empty-interface: Empty,
cond-class: [X?Y],
es-prior-interface: prior(X),
intensional-universe: IType,
tag-by: zT,
ldag: LabeledDAG(T),
labeled-graph: LabeledGraph(T),
fset: FSet{T},
isect2: T1 T2,
b-union: A 
B,
bag: bag(T),
axiom: Ax,
IdLnk: IdLnk,
append: as @ bs,
locl: locl(a),
Knd: Knd,
eq_knd: a = b,
so_apply: x[s1;s2],
Subst': Error :Subst',
AssertBY: Error :AssertBY,
Repeat: Error :Repeat,
int_seg: {i..j},
divides: b | a,
assoced: a ~ b,
set_leq: a b,
set_lt: a <p b,
grp_lt: a < b,
l_contains: A B,
inject: Inj(A;B;f),
reducible: reducible(a),
prime: prime(a),
l_exists: (xL. P[x]),
fun-connected: y is f*(x),
qle: r s,
qless: r < s,
q-rel: q-rel(r;x),
i-finite: i-finite(I),
i-closed: i-closed(I),
fset-member: a s,
f-subset: xs ys,
fset-closed: (s closed under fs),
l_disjoint: l_disjoint(T;l1;l2),
cs-not-completed: in state s, a has not completed inning i,
cs-archived: by state s, a archived v in inning i,
cs-passed: by state s, a passed inning i without archiving a value,
cs-archive-blocked: in state s, ws' blocks ws from archiving v in inning i,
cs-precondition: state s may consider v in inning i,
cs-inning-committed: in state s, inning i has committed v,
cs-inning-committable: in state s, inning i could commit v
Lemmas :
bfalse_wf,
es-collect-filter_functionality,
es-collect-filter_wf,
mapfilter-class_wf,
E-prior-class-when,
prior-imax-class-lb2,
uall_wf,
imax-class_wf,
assert_of_le_int,
iff_functionality_wrt_iff,
l_all_wf,
l_all_wf2,
not_functionality_wrt_iff,
iff_transitivity,
outl_wf,
isl_wf,
pair-prior-val,
es-interface-pair-prior_wf,
eq_int_wf,
paxos-state-value_wf,
assert_of_eq_int,
not_functionality_wrt_uiff,
is-pair-prior,
es-interface-subtype_rel,
es-collect-filter-max_wf,
collect-filter-max-map-class,
paxos-state-ballot_wf,
btrue_wf,
es-collect-filter-max_functionality,
es-interface-val_wf2,
intensional-universe_wf,
es-interface-at_wf,
member-interface-at,
E-map-class,
sq_stable__assert,
is-prior-val,
is-max-fst,
es-le-not-locl,
decidable__es-locl,
decidable__es-le,
es-causle-le,
prior-val-val,
max-fst-property,
es-causl_wf,
es-locl_transitivity1,
es-causl_weakening,
max-fst-class_wf,
es-prior-val_wf,
bnot_wf,
assert_of_bnot,
eqff_to_assert,
uiff_transitivity,
eqtt_to_assert,
unit_subtype_base,
int_subtype_base,
set_subtype_base,
product_subtype_base,
union_subtype_base,
es-E-interface_functionality,
paxos-state-reservation_wf,
paxos-state-name_wf,
paxos-state-info_wf,
bool_subtype_base,
assert_elim,
es-causle_wf,
l_member_wf,
squash_wf,
iff_weakening_uiff,
es-interface-extensionality,
true_wf,
atom2_subtype_base,
uiff_inversion,
in-eclass_wf,
rev_implies_wf,
es-loc_wf,
eclass-val_wf,
Paxos-spec8_wf,
nat_plus_wf,
Id_wf,
eclass_wf,
Paxos-spec7-body_wf,
map-class_wf,
nat_wf,
pi2_wf,
Paxos-spec7_wf,
event-ordering+_wf,
es-E_wf,
event-ordering+_inc,
paxos-acceptor-state_wf,
bool_wf,
nat_plus_inc,
unit_wf,
member_wf,
es-base-E_wf,
subtype_rel_self,
nat_plus_properties,
le_wf,
is-map-class,
subtype_rel_wf,
es-interface-top,
assert_wf,
map-class-val,
Paxos-spec8-locations,
es-collect-opt-max_wf,
pi1_wf_top,
top_wf,
alle-lt_wf,
es-first-at_wf,
uiff_wf,
is-collect-opt-max,
collect-opt-max-val,
es-filter-image_wf,
length_wf_nat,
ifthenelse_wf,
lt_int_wf,
nat_properties,
is-mapfilter-class,
mapfilter-class-val,
pi1_wf,
subtype_base_sq,
le_int_wf,
es-interface-accum_wf,
not_wf,
false_wf,
imax_wf,
imax_ub,
is-interface-accum,
es-interface-accum-val,
es-prior-class-when_wf,
es-interface-subtype_rel2,
es-tagged-true-class_wf,
is-tagged-true,
tagged-true-val,
max-f-class_wf,
accum-class_wf,
is-max-f-class,
max-f-class-val,
it_wf,
is-prior-class-when,
prior-class-when-val,
es-E-interface_wf,
es-le_wf,
es-locl_wf,
es-class-causal-mrel-fail_wf,
iff_wf,
es-class-causal-rel-fail_wf,
MaxVote_wf,
no_repeats_wf
\mforall{}[Info:Type]
\mforall{}es:EO+(Info)
\mforall{}[T:Type]
\mforall{}leader:\mBbbN{} {}\mrightarrow{} Id. \mforall{}failset:Id List. \mforall{}1a:EClass(Id \mtimes{} \mBbbN{}). \mforall{}X,Y:EClass(T).
(Paxos-spec8\{i:l\}(Info; es; T; leader; failset; 1a; X; Y)
{}\mRightarrow{} Paxos-spec7\{i:l\}(Info; es; T; leader; failset; (snd(p) where p from 1a); X; Y))
Date html generated:
2011_10_20-PM-04_49_13
Last ObjectModification:
2010_12_22-PM-06_39_52
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