Nuprl Lemma : Paxos-spec6-Proposals-increase
[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)].
[p1,p2:E(Proposal)]. (fst(Proposal(p1))) < (fst(Proposal(p2))) supposing (p1 <loc p2)
supposing 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-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-locl: (e <loc e'),
Id: Id,
bool: ,
nat_plus: ,
nat: ,
uimplies: b supposing a,
uall: [x:A]. B[x],
pi1: fst(t),
less_than: a < b,
function: x:A B[x],
product: x:A B[x],
list: type List,
int: ,
universe: Type
Definitions :
squash: 
T,
bag_only_single: bag_only_single{bag_only_single_compseq_tag_def:o}(x),
bag_size_single: bag_size_single{bag_size_single_compseq_tag_def:o}(x),
bag-only: only(bs),
bag-size: bag-size(bs),
bag_size_empty: bag_size_empty{bag_size_empty_compseq_tag_def:o},
limited-type: LimitedType,
bfalse: ff,
eq_bool: p =b q,
le_int: i 
z j,
eq_int: (i =
j),
null: null(as),
set_blt: a <
b,
grp_blt: a < b,
dcdr-to-bool: [d],
bl-all: (x
L.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-member: deq-member(eq;x;L),
deq-disjoint: deq-disjoint(eq;as;bs),
deq-all-disjoint: deq-all-disjoint(eq;ass;bs),
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,
bnot: 
b,
unit: Unit,
eclass-compose1: f o X,
grp_car: |g|,
cand: A c B,
guard: {T},
btrue: tt,
sq_type: SQType(T),
true: True,
real: 
,
rationals: 
,
atom: Atom,
es-base-E: es-base-E(es),
token: "$token",
bag: bag(T),
record-select: r.x,
infix_ap: x f y,
es-causl: (e < e'),
alle-lt: e<e'.P[e],
es-le: e loc e' ,
es-causle: e c e',
l_all: (xL.P[x]),
natural_number: $n,
add: n + m,
length: ||as||,
apply: f a,
es-loc: loc(e),
fpf: a:A fp-> B[a],
l_member: (x l),
rev_implies: P 
Q,
iff: P Q,
union: left + right,
or: P Q,
void: Void,
false: False,
eq_atom: x =a y,
eq_atom: eq_atom$n(x;y),
exists: x:A. B[x],
decide: case b of inl(x) => s[x] | inr(y) => t[y],
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,
implies: P Q,
no_repeats: no_repeats(T;l),
dep-isect: Error :dep-isect,
record+: record+,
strong-subtype: strong-subtype(A;B),
le: A B,
ge: i 
j ,
not: A,
uiff: uiff(P;Q),
paxos-acceptor-state: AcceptorState,
bool: ,
subtype_rel: A 
r B,
int: ,
set: {x:A| B[x]} ,
es-E: E,
subtype: S T,
all: x:A. B[x],
axiom: Ax,
prop: 
,
member: t T,
universe: Type,
nat_plus: ,
list: type List,
Id: Id,
function: x:A B[x],
so_lambda: 
x y.t[x; y],
uall: [x:A]. B[x],
so_lambda: x.t[x],
uimplies: b supposing a,
isect: 
x:A. B[x],
es-locl: (e <loc e'),
event-ordering+: EO+(Info),
event_ordering: EO,
es-E-interface: E(X),
top: Top,
Paxos-spec6-body: Paxos-spec6-body,
RepeatFor: Error :RepeatFor,
D: Error :D,
MaAuto: Error :MaAuto,
AssertBY: Error :AssertBY,
assert: 
b,
in-eclass: e X,
and: P Q,
less_than: a < b,
pi2: snd(t),
pi1: fst(t),
eclass-val: X(e),
CollapseTHEN: Error :CollapseTHEN,
CollapseTHENA: Error :CollapseTHENA,
equal: s = t,
eclass: EClass(A[eo; e]),
product: x:A B[x],
nat: ,
es-filter-image: f[X],
lambda: x.A[x],
spreadn: spread3,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
single-bag: {x},
pair: <a, b>,
empty-bag: {},
multiply: n * m,
it: 
,
paxos-state-reservation: Reservation(s),
es-interface-predecessors: (X)(e),
filter: filter(P;l),
es-first-at: e is first@ i s.t. e.P[e],
IdLnk: IdLnk,
proper-iseg: L1 < L2,
iseg: l1 l2,
l_exists: (xL. P[x]),
gt: i > j,
map: map(f;as),
eq_knd: a = b,
fpf-dom: x dom(f),
intensional-universe: IType,
nil: [],
list_ind: list_ind def,
int_eq: if a=b then c else d,
atom_eq: atomeqn def,
sqequal: s ~ t,
so_apply: x[s],
append: as @ bs,
locl: locl(a),
Knd: Knd,
atom: Atom$n,
id-deq: IdDeq,
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),
es-prior-val: (X)',
minus: -n,
mapfilter: mapfilter(f;P;L),
paxos-state-ballot: Ballot(s),
list-max: list-max(x.f[x];L),
spread: spread def,
let: let,
l_contains: A B,
inject: Inj(A;B;f),
reducible: reducible(a),
prime: prime(a),
fun-connected: y is f*(x),
qle: r s,
qless: r < s,
q-rel: q-rel(r;x),
sq_exists: x:{A| B[x]},
i-finite: i-finite(I),
i-closed: i-closed(I),
p-outcome: Outcome,
dstype: dstype(TypeNames; d; a),
fset-member: a s,
f-subset: xs ys,
fset: FSet{T},
fset-closed: (s closed under fs),
MaName: MaName,
l_disjoint: l_disjoint(T;l1;l2),
consensus-state3: consensus-state3(T),
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,
consensus-rcv: consensus-rcv(V;A),
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),
runEvents: runEvents(r),
decidable: Dec(P),
lg-edge: lg-edge(g;a;b),
int_nzero: 
Lemmas :
length_wf1,
es-locl_irreflexivity,
es-le_weakening_eq,
es-locl_transitivity2,
es-first-at-unique,
int_sq,
decidable__equal_int,
first-at-filter-interface-predecessors1,
alle-lt_wf,
set_subtype_base,
int_subtype_base,
es-loc_wf,
paxos-state-reservation_wf,
es-interface-predecessors_wf,
mapfilter_wf,
paxos-state-ballot_wf,
list-max_wf,
pos_length2,
length_wf_nat,
list-subtype,
l_member_wf,
uiff_wf,
assert-eq-id,
nat_properties,
btrue_wf,
bfalse_wf,
unit_wf,
intensional-universe_wf,
es-interface-val_wf2,
pos-length,
equal-nil-sq-nil,
filter_wf,
length-map,
filter_type,
list-max-property2,
es-prior-val_wf,
paxos-state-value_wf,
assert_of_eq_int,
not_functionality_wrt_uiff,
es-E_wf,
event-ordering+_inc,
es-locl_wf,
es-E-interface_wf,
Paxos-spec6-body_wf,
eclass_wf,
nat_wf,
Id_wf,
nat_plus_wf,
event-ordering+_wf,
uall_wf,
Paxos-spec6-Collect-invariant,
member_wf,
subtype_rel_wf,
es-interface-top,
nat_plus_inc,
bool_wf,
paxos-acceptor-state_wf,
es-base-E_wf,
subtype_rel_self,
nat_plus_properties,
le_wf,
assert_wf,
false_wf,
ifthenelse_wf,
in-eclass_wf,
true_wf,
subtype_base_sq,
bool_subtype_base,
assert_elim,
pi1_wf_top,
eclass-val_wf,
top_wf,
pi2_wf,
eqtt_to_assert,
not_wf,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
bnot_wf,
pi1_wf,
eq_int_wf,
bag-size_wf,
assert_of_lt_int,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
le_int_wf,
lt_int_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)].
\mforall{}[p1,p2:E(Proposal)]. (fst(Proposal(p1))) < (fst(Proposal(p2))) supposing (p1 <loc p2)
supposing 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_40_17
Last ObjectModification:
2011_06_18-PM-02_06_15
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