Nuprl Lemma : Paxos-spec7-body-spec6
[Info:Type]
es:EO+(Info). failset:Id List.
[T:Type]
f:
. acceptors:Id List. Reserve,NoProposal,NewBallot:EClass(). VoteState:EClass(AcceptorState).
Proposal:EClass( 
T). AcceptOrReject:EClass( T 
). leader: 
Id. Decide,Input:EClass(T).
Vote:EClass(Id ). Collect:EClass( 
T).
(Paxos-spec7-body{i:l}(Info;
es;
T;
f;
acceptors;
Reserve;
VoteState;
Proposal;
AcceptOrReject;
leader;
Decide;
Vote;
Input;
Collect;
NoProposal;
NewBallot;
failset)
Paxos-spec6-body{i:l}(Info;es;T;f;acceptors;
Reserve;VoteState;Proposal;
AcceptOrReject;leader;Decide;
Vote;Input;Collect;NoProposal;
NewBallot;failset))
Proof not projected
Definitions occuring in Statement :
Paxos-spec7-body: Paxos-spec7-body,
Paxos-spec6-body: Paxos-spec6-body,
paxos-acceptor-state: AcceptorState,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
Id: Id,
bool: ,
nat: ,
uall: [x:A]. B[x],
all: x:A. B[x],
implies: P Q,
function: x:A B[x],
product: x:A B[x],
list: type List,
int: ,
universe: Type
Definitions :
pair: <a, b>,
paxos-state-info: Info(s),
paxos-state-reservation: Reservation(s),
paxos-state-name: Name(s),
rev_implies: P 
Q,
in-eclass: e 
X,
bfalse: ff,
eclass-val: X(e),
pi1: fst(t),
pi2: snd(t),
apply: f a,
es-loc: loc(e),
es-tagged-true-class: Tagged_tt(X),
MaxVote: MaxVote(es;T;Vote;e;s),
iff: P Q,
es-first-at: e is first@ i s.t. e.P[e],
es-class-def: es-class-def,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
es-locl: (e <loc e'),
assert: 
b,
so_lambda: 
x.t[x],
alle-lt: e<e'.P[e],
es-le: e 
loc e' ,
union: left + right,
or: P 
Q,
top: Top,
less_than: a < b,
es-class-causal-mrel-fail: es-class-causal-mrel-fail,
paxos-acceptor-state: AcceptorState,
bool: ,
subtype: S 
T,
int: ,
event_ordering: EO,
es-E: E,
lambda: x.A[x],
member: t T,
prop: 
,
Paxos-spec7-body: Paxos-spec7-body,
so_lambda: x y.t[x; y],
eclass: EClass(A[eo; e]),
nat: ,
Id: Id,
list: type List,
Paxos-spec6-body: Paxos-spec6-body,
no_repeats: no_repeats(T;l),
uall: [x:A]. B[x],
uimplies: b supposing a,
isect: 
x:A. B[x],
le: A B,
not: 
A,
implies: P Q,
false: False,
void: Void,
equal: s = t,
es-class-causal-rel-fail: es-class-causal-rel-fail,
and: P 
Q,
exists: 
x:A. B[x],
product: x:A B[x],
es-E-interface: E(X),
set: {x:A| B[x]} ,
universe: Type,
function: x:A B[x],
event-ordering+: EO+(Info),
all: x:A. B[x],
RepeatFor: Error :RepeatFor,
Try: Error :Try,
CollapseTHEN: Error :CollapseTHEN,
D: Error :D,
ExRepD: Error :ExRepD,
Unfold: Error :Unfold,
Auto: Error :Auto,
CollapseTHENA: Error :CollapseTHENA,
tactic: Error :tactic,
decide: case b of inl(x) => s[x] | inr(y) => t[y],
true: True,
mapfilter: mapfilter(f;P;L),
es-interface-predecessors: (X)(e),
band: p q,
strong-subtype: strong-subtype(A;B),
ge: i 
j ,
uiff: uiff(P;Q),
subtype_rel: A r B,
real: 
,
rationals: 
,
nat_plus: 
,
dep-isect: Error :dep-isect,
eq_atom: x =a y,
eq_atom: eq_atom$n(x;y),
record+: record+,
bag: bag(T),
l_member: (x l),
es-causle: e c e',
id-deq: IdDeq,
deq-member: deq-member(eq;x;L),
filter: filter(P;l),
multiply: n * m,
length: ||as||,
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]),
es-interface-pair-prior: X;Y,
paxos-state-value: Value(s),
minus: -n,
eq_int: (i =
j),
ifthenelse: if b then t else f fi ,
spreadn: spread4,
map-class: (f[v] where v from X),
MaAuto: Error :MaAuto,
btrue: tt,
natural_number: $n,
add: n + m,
paxos-state-ballot: Ballot(s),
cand: A c B,
list-max: list-max(x.f[x];L),
spread: spread def,
let: let,
grp_car: |g|,
limited-type: LimitedType,
es-prior-val: (X)',
fpf: a:A fp-> B[a],
record-select: r.x,
THENL_v2: Error :THENL,
THENL_cons: Error :THENL_cons,
THENL_cons: Error :THENL_nil,
fpf-dom: x dom(f),
eq_knd: a = b,
guard: {T},
so_apply: x[s],
cond-class: [X?Y],
axiom: Ax,
es-causl: (e < e'),
infix_ap: x f y,
es-local-pred: last(P),
sq_stable: SqStable(P),
squash: 
T,
fpf-sub: f g,
modulus-of-ccontinuity: modulus-of-ccontinuity(omega;I;f),
partitions: partitions(I;p),
i-member: r I,
rleq: x y,
rnonneg: rnonneg(r),
req: x = y,
is_accum_splitting: is_accum_splitting(T;A;L;LL;L2;f;g;x),
is_list_splitting: is_list_splitting(T;L;LL;L2;f),
token: "$token",
record: record(x.T[x]),
fpf-cap: f(x)?z,
b-union: A 
B,
isect2: T1 T2,
fset: FSet{T},
labeled-graph: LabeledGraph(T),
ldag: LabeledDAG(T),
tag-by: zT,
intensional-universe: IType,
es-interface-at: X@i,
es-prior-interface: prior(X),
es-pred: pred(e),
IdLnk: IdLnk,
Knd: Knd,
es-base-E: es-base-E(es),
atom: Atom,
sq_type: SQType(T),
atom: Atom$n,
locl: locl(a),
append: as @ bs,
sqequal: s ~ t,
atom_eq: atomeqn def,
can-apply: can-apply(f;x),
isl: isl(x),
int_eq: if a=b then c else d,
unit: Unit,
inl: inl x ,
inr: inr x ,
decidable: Dec(P),
ses-legal-sequence: Legal(pas) given prvt,
ses-action: Action(e),
lg-edge: lg-edge(g;a;b),
path-goes-thru: x-f*-y thru i,
cut-order: a (X;f) b,
collect-event: collect-event(es;X;n;v.num[v];L.P[L];e),
same-thread: same-thread(es;p;e;e'),
es-r-immediate-pred: es-r-immediate-pred(es;R;e';e),
es-fset-loc: i locs(s),
existse-between3: e(e1,e2].P[e],
existse-between2: e[e1,e2].P[e],
alle-between2: e[e1,e2].P[e],
existse-between1: e[e1,e2).P[e],
alle-between1: e[e1,e2).P[e],
alle-le: ee'.P[e],
existse-le: ee'.P[e],
existse-before: e<e'.P[e],
cs-precondition: state s may consider v in inning i,
cs-archive-blocked: in state s, ws' blocks ws from archiving v in inning i,
cs-inning-committable: in state s, inning i could commit v ,
cs-inning-committed: in state s, inning i has committed v,
cs-passed: by state s, a passed inning i without archiving a value,
cs-archived: by state s, a archived v in inning i,
cs-not-completed: in state s, a has not completed inning i,
l_disjoint: l_disjoint(T;l1;l2),
fset-closed: (s closed under fs),
f-subset: xs ys,
fset-member: a s,
p-outcome: Outcome,
i-closed: i-closed(I),
i-finite: i-finite(I),
q-rel: q-rel(r;x),
qless: r < s,
qle: r s,
fun-connected: y is f*(x),
l_all: (xL.P[x]),
l_exists: (xL. P[x]),
prime: prime(a),
reducible: reducible(a),
inject: Inj(A;B;f),
l_contains: A B,
grp_lt: a < b,
set_lt: a <p b,
set_leq: a b,
assoced: a ~ b,
divides: b | a,
es-init: es-init(es;e),
ldst: destination(l),
lsrc: source(l),
null: null(as),
es-first: first(e),
map: map(f;as),
value-type: value-type(T),
prime_ideal_p: IsPrimeIdeal(R;P),
integ_dom_p: IsIntegDom(r),
grp_leq: a b,
monoid_hom_p: IsMonHom{M1,M2}(f),
group_p: IsGroup(T;op;id;inv),
monoid_p: IsMonoid(T;op;id),
monot: monot(T;x,y.R[x; y];f),
cancel: Cancel(T;S;op),
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
fun_thru_1op: fun_thru_1op(A;B;opa;opb;f),
dist_1op_2op_lr: Dist1op2opLR(A;1op;2op),
action_p: IsAction(A;x;e;S;f),
bilinear_p: IsBilinear(A;B;C;+a;+b;+c;f),
bilinear: BiLinear(T;pl;tm),
inverse: Inverse(T;op;id;inv),
comm: Comm(T;op),
assoc: Assoc(T;op),
ident: Ident(T;op;id),
coprime: CoPrime(a,b),
uconnex: uconnex(T; x,y.R[x; y]),
connex: Connex(T;x,y.R[x; y]),
uanti_sym: UniformlyAntiSym(T;x,y.R[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y]),
utrans: UniformlyTrans(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
usym: UniformlySym(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
urefl: UniformlyRefl(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
eqfun_p: IsEqFun(T;eq),
inv_funs: InvFuns(A;B;f;g),
uni_sat: a = !x:T. Q[x],
rev_uimplies: rev_uimplies(P;Q),
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',
bnot: b,
bimplies: p q,
bor: p q,
THENM: Error :THENM,
ParallelOp: Error :ParallelOp,
skip: Error :skip,
quotient: x,y:A//B[x; y],
lt_int: i <z j,
le_int: i z j,
gt: i > j,
iseg: l1 l2,
proper-iseg: L1 < L2,
combination: Combination(n;T),
listp: A List,
deq: EqDecider(T),
nil: [],
cons: [car / cdr],
imax-list: imax-list(L),
compose: f o g,
paxos-decide-def: paxos-decide-def(es;f;leader;T;Decide;Proposal;Vote),
it: 
,
spreadn: spread3,
es-filter-image: f[X],
es-collect-filter: Collect(size v's from X with maximum num[v] such that P[v]),
mapfilter-class: (f[v] where v from X such that P[v]),
eq_bool: p =b q,
collect_accum: collect_accum(x.num[x];init;a,v.f[a; v];a.P[a]),
collect_filter: collect_filter(),
es-interface-accum: es-interface-accum(f;x;X),
es-collect-accum: es-collect-accum(X;x.num[x];init;a,v.f[a; v];a.P[a]),
paxos-decide-relation: paxos-decide-relation(es; f; leader; Proposal; Vote; d; b)
Lemmas :
mapfilter-class-val,
paxos-decide-relation_wf,
assert_of_band,
and_functionality_wrt_uiff,
assert_of_eq_bool,
eq_bool_tt,
es-interface-val_wf,
assert_functionality_wrt_uiff,
es-collect-filter-val,
is-collect-filter,
is-mapfilter-class,
eq_bool_wf,
mapfilter-class_wf,
pi2_wf,
es-collect-filter_wf,
paxos-decide-def_wf,
map-map,
member_null,
pi1_wf_top,
list-max-imax-list,
pi1_wf,
imax-list_wf,
l_all_wf2,
l_all_wf,
deq_wf,
filter_wf_top,
id-deq_wf,
deq-member_wf,
map_length,
non_neg_length,
list-max-property,
map_wf,
equal-nil-sq-nil,
pos-length,
set_subtype_base,
list_subtype_base,
map-class-val,
eqtt_to_assert,
eqff_to_assert,
assert_of_bnot,
not_functionality_wrt_uiff,
bnot_wf,
pair-prior-val,
nat_plus_wf,
uiff_wf,
length-map,
assert_of_eq_int,
pos_length2,
first-at-filter-interface-predecessors1,
band_tt_simp,
list-equal-set,
sq_stable__equal,
sq_stable_wf,
list-equal-set2,
squash_wf,
event_ordering_wf,
int_subtype_base,
es-loc-pred,
decidable__assert,
decidable_wf,
es-interface-subtype_rel2,
es-interface-val_wf2,
unit_wf,
bfalse_wf,
band_wf,
iff_weakening_uiff,
is-map-class,
uiff_transitivity,
is-pair-prior,
and_functionality_wrt_uiff3,
map-class_wf,
es-interface-pair-prior_wf,
es-collect-filter-max_wf,
assert_elim,
bool_subtype_base,
subtype_base_sq,
assert-eq-id,
filter_type,
l_member_wf,
list-subtype,
subtype_rel_self,
es-base-E_wf,
es-causl_wf,
is-prior-val,
intensional-universe_wf,
sq_stable__assert,
uiff_inversion,
rev_implies_wf,
length_wf_nat,
assert_witness,
in-eclass_wf,
member_wf,
subtype_rel_wf,
es-interface-top,
false_wf,
ifthenelse_wf,
true_wf,
es-interface-predecessors_wf,
filter_wf,
length_wf1,
es-loc_wf,
eq_int_wf,
eclass-val_wf,
es-prior-val_wf,
top_wf,
mapfilter_wf,
list-max_wf,
paxos-state-value_wf,
is-collect-filter-max,
collect-filter-max-val,
paxos-state-ballot_wf,
nat_properties,
paxos-state-reservation_wf,
btrue_wf,
event-ordering+_inc,
es-E_wf,
nat_wf,
event-ordering+_wf,
eclass_wf,
paxos-acceptor-state_wf,
bool_wf,
Id_wf,
Paxos-spec7-body_wf,
Paxos-spec6-body_wf,
not_wf,
es-E-interface_wf,
es-le_wf,
assert_wf,
alle-lt_wf,
es-locl_wf,
es-class-causal-mrel-fail_wf,
le_wf,
es-first-at_wf,
es-class-def_wf,
iff_wf,
es-class-causal-rel-fail_wf,
es-tagged-true-class_wf,
MaxVote_wf,
no_repeats_wf
\mforall{}[Info:Type]
\mforall{}es:EO+(Info). \mforall{}failset:Id List.
\mforall{}[T:Type]
\mforall{}f:\mBbbN{}. \mforall{}acceptors:Id List. \mforall{}Reserve,NoProposal,NewBallot:EClass(\mBbbN{}).
\mforall{}VoteState:EClass(AcceptorState). \mforall{}Proposal:EClass(\mBbbN{} \mtimes{} T). \mforall{}AcceptOrReject:EClass(\mBbbN{} \mtimes{} T \mtimes{} \mBbbB{}).
\mforall{}leader:\mBbbN{} {}\mrightarrow{} Id. \mforall{}Decide,Input:EClass(T). \mforall{}Vote:EClass(Id \mtimes{} \mBbbN{} \mtimes{} \mBbbB{}).
\mforall{}Collect:EClass(\mBbbN{} \mtimes{} \mBbbZ{} \mtimes{} T).
(Paxos-spec7-body\{i:l\}(Info;
es;
T;
f;
acceptors;
Reserve;
VoteState;
Proposal;
AcceptOrReject;
leader;
Decide;
Vote;
Input;
Collect;
NoProposal;
NewBallot;
failset)
{}\mRightarrow{} Paxos-spec6-body\{i:l\}(Info;es;T;f;acceptors;
Reserve;VoteState;Proposal;
AcceptOrReject;leader;Decide;
Vote;Input;Collect;NoProposal;
NewBallot;failset))
Date html generated:
2011_10_20-PM-04_44_47
Last ObjectModification:
2011_06_18-PM-02_10_25
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