{ 
[Info,A1,A2:Type]. [X1:EClass(A1)]. [X2:EClass(A2)]. [size:
].
[num1:A1 
]. [num2:A2 ]. [P1:A1 
]. [P2:A2 ].
Collect(size v's from X1 with maximum num1[v] such that P1[v])
= Collect(size v's from X2 with maximum num2[v] such that P2[v])
supposing es:EO+(Info). e:E.
((
e 
X1 
e X2)
((e X1)
(e X2)
((num1[X1(e)] = num2[X2(e)])
(P1[X1(e)] P2[X2(e)])))) }
{ Proof }
Definitions occuring in Statement :
es-collect-filter: Collect(size v's from X with maximum num[v] such that P[v]),
eclass-val: X(e),
in-eclass: e X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-E: E,
assert: b,
bool: ,
nat_plus: ,
nat: ,
uimplies: b supposing a,
uall: [x:A]. B[x],
so_apply: x[s],
all: x:A. B[x],
iff: P Q,
implies: P Q,
and: P Q,
function: x:A B[x],
universe: Type,
equal: s = t
Definitions :
universe: Type,
member: t T,
equal: s = t,
function: x:A B[x],
all: x:A. B[x],
event-ordering+: EO+(Info),
dep-isect: Error :dep-isect,
event_ordering: EO,
es-E: E,
top: Top,
token: "$token",
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
atom: Atom,
subtype_rel: A 
r B,
eq_atom: eq_atom$n(x;y),
apply: f a,
record-select: r.x,
record+: record+,
lambda: 
x.A[x],
so_lambda: 
x y.t[x; y],
eclass: EClass(A[eo; e]),
union: left + right,
nat_plus: ,
nat: ,
set: {x:A| B[x]} ,
bool: ,
prop: 
,
product: x:A 
B[x],
and: P Q,
iff: P Q,
assert: b,
eclass-val: X(e),
so_apply: x[s],
subtype: S T,
implies: P Q,
es-E-interface: E(X),
rev_implies: P Q,
cand: A c B,
in-eclass: e X,
so_lambda: x.t[x],
es-collect-filter: Collect(size v's from X with maximum num[v] such that P[v]),
pair: <a, b>,
int: 
,
fpf: a:A fp-> B[a],
strong-subtype: strong-subtype(A;B),
decide: case b of inl(x) => s[x] | inr(y) => t[y],
fpf-dom: x dom(f),
less_than: a < b,
or: P 
Q,
Knd: Knd,
IdLnk: IdLnk,
Id: Id,
l_member: (x l),
sq_stable: SqStable(P),
guard: {T},
eq_knd: a = b,
list: type List,
isect: 
x:A. B[x],
es-first-at: e is first@ i s.t. e.P[e],
alle-lt: e<e'.P[e],
not: 
A,
length: ||as||,
es-loc: loc(e),
le: A 
B,
es-interface-predecessors: (X)(e),
eq_int: (i =
j),
band: p q,
filter: filter(P;l),
void: Void,
false: False,
es-interface-at: X@i,
es-collect-accum: es-collect-accum(X;x.num[x];init;a,v.f[a; v];a.P[a]),
es-interface-accum: es-interface-accum(f;x;X),
collect_filter: collect_filter(),
collect_accum: collect_accum(x.num[x];init;a,v.f[a; v];a.P[a]),
cond-class: [X?Y],
ge: i 
j ,
sq_type: SQType(T),
isl: isl(x),
can-apply: can-apply(f;x),
es-locl: (e <loc e'),
true: True,
real: 
,
grp_car: |g|,
squash: 
T,
btrue: tt,
infix_ap: x f y,
es-causl: (e < e'),
decidable: Dec(P),
exists: 
x:A. B[x],
divides: b | a,
assoced: a ~ b,
set_leq: a b,
set_lt: a <p b,
grp_lt: a < b,
l_all: (xL.P[x]),
l_exists: (xL. P[x]),
l_contains: A B,
inject: Inj(A;B;f),
reducible: reducible(a),
prime: prime(a),
l_disjoint: l_disjoint(T;l1;l2),
fun-connected: y is f*(x),
qle: r s,
qless: r < s,
q-rel: q-rel(r;x),
p-outcome: Outcome,
fset-member: a s,
f-subset: xs ys,
fset-closed: (s closed under fs),
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,
es-le: e loc e' ,
es-causle: e c 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),
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,
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]),
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y]),
connex: Connex(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),
no_repeats: no_repeats(T;l),
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),
fpf-sub: f g,
limited-type: LimitedType,
append: as @ bs,
locl: locl(a),
natural_number: $n,
imax: imax(a;b),
MaAuto: Error :MaAuto,
CollapseTHEN: Error :CollapseTHEN,
D: Error :D,
THENM: Error :THENM,
CollapseTHENA: Error :CollapseTHENA,
Complete: Error :Complete,
Try: Error :Try,
ParallelOp: Error :ParallelOp,
RepeatFor: Error :RepeatFor
Lemmas :
false_wf,
sq_stable_from_decidable,
decidable__assert,
assert_elim,
int_sq,
length_wf1,
es-loc_wf,
filter_wf,
es-interface-predecessors_wf,
es-E-interface_wf,
band_wf,
eq_int_wf,
squash_wf,
true_wf,
iff_imp_equal_bool,
es-locl_wf,
es-interface-predecessors-equal-subtype,
length_wf_nat,
es-interface-extensionality,
Id_wf,
not_wf,
es-first-at_wf,
alle-lt_wf,
es-interface-subtype_rel2,
top_wf,
rev_implies_wf,
es-collect-filter-val,
is-collect-filter,
le_wf,
member_wf,
subtype_rel_wf,
es-interface-top,
es-collect-filter_wf,
in-eclass_wf,
iff_wf,
assert_wf,
eclass-val_wf,
event-ordering+_inc,
bool_wf,
nat_wf,
nat_plus_wf,
eclass_wf,
es-E_wf,
subtype_rel_self,
event-ordering+_wf
\mforall{}[Info,A1,A2:Type]. \mforall{}[X1:EClass(A1)]. \mforall{}[X2:EClass(A2)]. \mforall{}[size:\mBbbN{}\msupplus{}]. \mforall{}[num1:A1 {}\mrightarrow{} \mBbbN{}].
\mforall{}[num2:A2 {}\mrightarrow{} \mBbbN{}]. \mforall{}[P1:A1 {}\mrightarrow{} \mBbbB{}]. \mforall{}[P2:A2 {}\mrightarrow{} \mBbbB{}].
Collect(size v's from X1 with maximum num1[v] such that P1[v])
= Collect(size v's from X2 with maximum num2[v] such that P2[v])
supposing \mforall{}es:EO+(Info). \mforall{}e:E.
((\muparrow{}e \mmember{}\msubb{} X1 \mLeftarrow{}{}\mRightarrow{} \muparrow{}e \mmember{}\msubb{} X2)
\mwedge{} ((\muparrow{}e \mmember{}\msubb{} X1)
{}\mRightarrow{} (\muparrow{}e \mmember{}\msubb{} X2)
{}\mRightarrow{} ((num1[X1(e)] = num2[X2(e)]) \mwedge{} (\muparrow{}P1[X1(e)] \mLeftarrow{}{}\mRightarrow{} \muparrow{}P2[X2(e)]))))
Date html generated:
2011_08_16-PM-06_11_08
Last ObjectModification:
2010_11_07-AM-02_18_10
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