Nuprl Lemma : Paxos-spec6-Collect-invariant
[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-spec6-body{i:l}(Info;es;T;f;acceptors;
Reserve;VoteState;Proposal;
AcceptOrReject;leader;Decide;
Vote;Input;Collect;NoProposal;
NewBallot;failset)
(c:E(Collect)
((loc(c) = (leader (fst(Collect(c)))))
(
L:{a:Id| (a 
acceptors)} List
((||L|| = (f + 1))
no_repeats(Id;L)
(aL.r:E(Reserve). ((loc(r) = a) r c
c ((fst(Collect(c))) = Reserve(r))))))
(nb:E(NewBallot). ((nb <loc c) ((fst(Collect(c))) = NewBallot(nb)))))))
Proof not projected
Definitions occuring in Statement :
Paxos-spec6-body: Paxos-spec6-body,
paxos-acceptor-state: AcceptorState,
es-E-interface: E(X),
eclass-val: X(e),
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-causle: e c e',
es-locl: (e <loc e'),
es-loc: loc(e),
Id: Id,
length: ||as||,
bool: ,
nat_plus: ,
nat: ,
uall: [x:A]. B[x],
pi1: fst(t),
all: x:A. B[x],
exists: x:A. B[x],
implies: P Q,
and: P Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A B[x],
product: x:A B[x],
list: type List,
add: n + m,
natural_number: $n,
int: ,
universe: Type,
equal: s = t,
l_all: (xL.P[x]),
no_repeats: no_repeats(T;l),
l_member: (x l)
Definitions :
null: null(as),
set_blt: a <
b,
grp_blt: a < b,
dcdr-to-bool: [d],
bl-all: (xL.P[x])_b,
bl-exists: (xL.P[x])_b,
b-exists: (i<n.P[i])_b,
eq_type: eq_type(T;T'),
qeq: qeq(r;s),
q_less: q_less(r;s),
q_le: q_le(r;s),
deq-disjoint: deq-disjoint(eq;as;bs),
deq-all-disjoint: deq-all-disjoint(eq;ass;bs),
eq_str: Error :eq_str,
eq_id: a = b,
eq_lnk: a = b,
es-eq-E: e = e',
es-bless: e <loc e',
es-ble: e loc e',
bimplies: p q,
bor: p 
q,
IdLnk: IdLnk,
limited-type: LimitedType,
eq_knd: a = b,
fpf-dom: x dom(f),
unit: Unit,
int_eq: if a=b then c else d,
intensional-universe: IType,
isl: isl(x),
can-apply: can-apply(f;x),
atom_eq: atomeqn def,
so_apply: x[s],
append: as @ bs,
locl: locl(a),
Knd: Knd,
atom: Atom$n,
record: record(x.T[x]),
sqequal: s ~ t,
id-deq: IdDeq,
deq-member: deq-member(eq;x;L),
paxos-state-info: Info(s),
es-tagged-true-class: Tagged_tt(X),
MaxVote: MaxVote(es;T;Vote;e;s),
paxos-state-name: Name(s),
paxos-state-value: Value(s),
minus: -n,
paxos-state-ballot: Ballot(s),
list-max: list-max(x.f[x];L),
spread: spread def,
eq_bool: p =b q,
band: p q,
bnot: 
b,
bfalse: ff,
pi2: snd(t),
pi1: fst(t),
es-prior-val: (X)',
fpf: a:A fp-> B[a],
strong-subtype: strong-subtype(A;B),
infix_ap: x f y,
es-causl: (e < e'),
grp_car: |g|,
filter: filter(P;l),
axiom: Ax,
es-loc: loc(e),
let: let,
es-first-at: e is first@ i s.t. e.P[e],
guard: {T},
lt_int: i <z j,
inr: inr x ,
inl: inl x ,
le_int: i z j,
spreadn: spread3,
es-filter-image: f[X],
btrue: tt,
sq_type: SQType(T),
ge: i 
j ,
uiff: uiff(P;Q),
true: True,
multiply: n * m,
add: n + m,
cand: A c B,
decide: case b of inl(x) => s[x] | inr(y) => t[y],
uimplies: b supposing a,
alle-lt: e<e'.P[e],
es-le: e loc e' ,
pair: <a, b>,
or: P Q,
rev_implies: P 
Q,
iff: P Q,
es-class-causal-mrel-fail: es-class-causal-mrel-fail,
es-class-def: es-class-def,
es-class-causal-rel-fail: es-class-causal-rel-fail,
void: Void,
false: False,
not: A,
le: A B,
real: 
,
rationals: 
,
prop: 
,
bool: ,
paxos-acceptor-state: AcceptorState,
union: left + right,
subtype: S 
T,
subtype_rel: A r B,
atom: Atom,
apply: f a,
es-base-E: es-base-E(es),
token: "$token",
ifthenelse: if b then t else f fi ,
record-select: r.x,
es-E: E,
dep-isect: Error :dep-isect,
eq_atom: x =a y,
eq_atom: eq_atom$n(x;y),
record+: record+,
member: t T,
universe: Type,
nat_plus: ,
so_lambda: 
x y.t[x; y],
eclass: EClass(A[eo; e]),
Paxos-spec6-body: Paxos-spec6-body,
l_member: (x l),
list: type List,
no_repeats: no_repeats(T;l),
Id: Id,
es-causle: e c e',
l_all: (xL.P[x]),
int: ,
set: {x:A| B[x]} ,
top: Top,
es-E-interface: E(X),
event-ordering+: EO+(Info),
event_ordering: EO,
es-locl: (e <loc e'),
nat: ,
equal: s = t,
exists: x:A. B[x],
isect: 
x:A. B[x],
uall: [x:A]. B[x],
so_lambda: x.t[x],
implies: P Q,
all: x:A. B[x],
function: x:A B[x],
product: x:A B[x],
and: P Q,
MaAuto: Error :MaAuto,
CollapseTHEN: Error :CollapseTHEN,
RepeatFor: Error :RepeatFor,
CollapseTHENA: Error :CollapseTHENA,
Auto: Error :Auto,
Complete: Error :Complete,
D: Error :D,
it: 
,
THENM: Error :THENM,
Unfold: Error :Unfold,
es-interface-predecessors: (X)(e),
eclass-val: X(e),
paxos-state-reservation: Reservation(s),
eq_int: (i =
j),
lambda: x.A[x],
mapfilter: mapfilter(f;P;L),
length: ||as||,
natural_number: $n,
less_than: a < b,
AssertBY: Error :AssertBY,
RepUR: Error :RepUR,
ExRepD: Error :ExRepD,
in-eclass: e X,
assert: 
b,
squash: 
T,
nil: [],
inject: Inj(A;B;f),
firstn: firstn(n;as),
cons: [car / cdr],
int_seg: {i..j
},
remove-repeats: remove-repeats(eq;L),
select: l[i],
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),
es-init: es-init(es;e),
es-pred: pred(e),
es-fix: f**(e),
es-p-le: e p e',
es-p-locl: e p< e',
causal-predecessor: causal-predecessor(es;p),
uni_sat: a = !x:T. Q[x],
inv_funs: InvFuns(A;B;f;g),
eqfun_p: IsEqFun(T;eq),
refl: Refl(T;x,y.E[x; y]),
urefl: UniformlyRefl(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
usym: UniformlySym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
utrans: UniformlyTrans(T;x,y.E[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y]),
uanti_sym: UniformlyAntiSym(T;x,y.R[x; y]),
connex: Connex(T;x,y.R[x; y]),
uconnex: uconnex(T; x,y.R[x; y]),
coprime: CoPrime(a,b),
ident: Ident(T;op;id),
assoc: Assoc(T;op),
comm: Comm(T;op),
inverse: Inverse(T;op;id;inv),
bilinear: BiLinear(T;pl;tm),
bilinear_p: IsBilinear(A;B;C;+a;+b;+c;f),
action_p: IsAction(A;x;e;S;f),
dist_1op_2op_lr: Dist1op2opLR(A;1op;2op),
fun_thru_1op: fun_thru_1op(A;B;opa;opb;f),
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
cancel: Cancel(T;S;op),
monot: monot(T;x,y.R[x; y];f),
monoid_p: IsMonoid(T;op;id),
group_p: IsGroup(T;op;id;inv),
monoid_hom_p: IsMonHom{M1,M2}(f),
grp_leq: a b,
integ_dom_p: IsIntegDom(r),
prime_ideal_p: IsPrimeIdeal(R;P),
value-type: value-type(T),
!hyp_hide: x,
es-before: before(e),
es-le-before: loc(e),
map: map(f;as),
hd: hd(l),
last: last(L),
MaName: MaName,
consensus-state3: consensus-state3(T),
consensus-rcv: consensus-rcv(V;A),
runEvents: runEvents(r),
divides: b | a,
assoced: a ~ b,
set_leq: a b,
set_lt: a <p b,
grp_lt: a < b,
l_contains: A B,
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),
p-outcome: Outcome,
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-inning-committed: in state s, inning i has committed v,
cs-inning-committable: in state s, inning i could commit v ,
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,
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,
lg-edge: lg-edge(g;a;b),
ses-action: Action(e),
ses-legal-sequence: Legal(pas) given prvt,
decidable: Dec(P),
tl: tl(l),
tag-by: zT,
ldag: LabeledDAG(T),
labeled-graph: LabeledGraph(T),
fset: FSet{T},
isect2: T1 T2,
b-union: A 
B,
fpf-cap: f(x)?z,
imax-class: (maximum f[v] lb with v from X),
es-prior-class-when: (X'?d) when Y,
map-class: (f[v] where v from X),
es-interface-at: X@i
Lemmas :
es-causl_transitivity1,
atom2_subtype_base,
l_all_cons,
no_repeats_cons,
and_functionality_wrt_uiff2,
decidable_wf,
decidable__not,
decidable__l_member,
decidable__equal_Id,
cons_member,
member-interface-predecessors,
l_member-settype,
es-causle_weakening_locl,
es-causle_transitivity,
set_subtype_base,
member-mapfilter,
property-from-l_member,
sq_stable_wf,
sq_stable__equal,
es-causl_wf,
es-le-causle,
es-le_wf,
es-causle-le,
sq_stable__assert,
member_filter,
select_wf,
es-interface-predecessors-no_repeats,
no_repeats_filter,
no_repeats_map,
no_repeats-settype,
inject_wf,
list-set-type2,
es-E-interface-subtype_rel,
l_all_wf2,
paxos-state-name_wf,
int_subtype_base,
pi1_wf,
pi1_wf_top,
uall_wf,
nat_plus_wf,
Id_wf,
eclass_wf,
nat_wf,
Paxos-spec6-body_wf,
es-E-interface_wf,
l_member_wf,
no_repeats_wf,
l_all_wf,
es-causle_wf,
es-locl_wf,
event-ordering+_wf,
es-E_wf,
es-base-E_wf,
subtype_rel_self,
event-ordering+_inc,
paxos-acceptor-state_wf,
bool_wf,
nat_plus_properties,
le_wf,
nat_plus_inc,
member_wf,
es-interface-top,
assert_wf,
false_wf,
ifthenelse_wf,
in-eclass_wf,
true_wf,
subtype_base_sq,
bool_subtype_base,
assert_elim,
length_wf_nat,
assert_witness,
es-interface-predecessors_wf,
filter_wf,
length_wf1,
es-loc_wf,
eq_int_wf,
eclass-val_wf,
paxos-state-reservation_wf,
subtype_rel_wf,
es-prior-val_wf,
top_wf,
length-map,
list-subtype,
filter_type,
iff_weakening_uiff,
uiff_inversion,
assert-eq-id,
nat_properties,
intensional-universe_wf,
mapfilter_wf,
list-max_wf,
paxos-state-ballot_wf,
btrue_wf,
bfalse_wf,
unit_wf,
es-interface-val_wf2,
list-max-property2,
paxos-state-value_wf,
not_wf,
no_repeats_witness,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
assert_of_bnot,
not_functionality_wrt_uiff,
bnot_wf
\mforall{}[Info:Type]
\mforall{}es:EO+(Info). \mforall{}failset:Id List.
\mforall{}[T:Type]
\mforall{}f:\mBbbN{}\msupplus{}. \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-spec6-body\{i:l\}(Info;es;T;f;acceptors;
Reserve;VoteState;Proposal;
AcceptOrReject;leader;Decide;
Vote;Input;Collect;NoProposal;
NewBallot;failset)
{}\mRightarrow{} (\mforall{}c:E(Collect)
((loc(c) = (leader (fst(Collect(c)))))
\mwedge{} (\mexists{}L:\{a:Id| (a \mmember{} acceptors)\} List
((||L|| = (f + 1))
\mwedge{} no\_repeats(Id;L)
\mwedge{} (\mforall{}a\mmember{}L.\mexists{}r:E(Reserve)
((loc(r) = a) \mwedge{} r c\mleq{} c \mwedge{} ((fst(Collect(c))) = Reserve(r))))))
\mwedge{} (\mexists{}nb:E(NewBallot). ((nb <loc c) \mwedge{} ((fst(Collect(c))) = NewBallot(nb)))))))
Date html generated:
2011_10_20-PM-04_37_38
Last ObjectModification:
2011_06_18-PM-02_02_47
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