Nuprl Lemma : bag-combine-filter
∀[A,B:Type]. ∀[p:A ⟶ 𝔹]. ∀[f:{a:A| ↑p[a]} ⟶ bag(B)]. ∀[ba:bag(A)].
(⋃a∈[a∈ba|p[a]].f[a] = ⋃a∈ba.if p[a] then f[a] else {} fi ∈ bag(B))
Proof
Definitions occuring in Statement :
bag-combine: ⋃x∈bs.f[x],
bag-filter: [x∈b|p[x]],
empty-bag: {},
bag: bag(T),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
cand: A c∧ B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
bag-filter: [x∈b|p[x]],
empty-bag: {},
bag-combine: ⋃x∈bs.f[x],
bag-map: bag-map(f;bs),
bag-union: bag-union(bbs),
concat: concat(ll),
nat: ℕ,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
decidable: Dec(P),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
bag-append: as + bs,
true: True
Lemmas referenced :
bag_wf,
list_wf,
permutation_wf,
equal_wf,
equal-wf-base,
assert_wf,
bool_wf,
quotient-member-eq,
permutation-equiv,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
filter_nil_lemma,
map_nil_lemma,
reduce_nil_lemma,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
filter_cons_lemma,
map_cons_lemma,
reduce_cons_lemma,
empty-bag_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
list_ind_nil_lemma,
bag-append_wf,
squash_wf,
true_wf,
bag-combine_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
because_Cache,
rename,
dependent_functionElimination,
independent_functionElimination,
productEquality,
isect_memberEquality,
axiomEquality,
functionEquality,
setEquality,
applyEquality,
functionExtensionality,
lambdaEquality,
independent_isectElimination,
independent_pairFormation,
hyp_replacement,
applyLambdaEquality,
setElimination,
intWeakElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
computeAll,
unionElimination,
promote_hyp,
hypothesis_subsumption,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
imageElimination,
equalityElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[p:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:\{a:A| \muparrow{}p[a]\} {}\mrightarrow{} bag(B)]. \mforall{}[ba:bag(A)].
(\mcup{}a\mmember{}[a\mmember{}ba|p[a]].f[a] = \mcup{}a\mmember{}ba.if p[a] then f[a] else \{\} fi )
Date html generated:
2017_10_01-AM-08_47_29
Last ObjectModification:
2017_07_26-PM-04_31_59
Theory : bags
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