Nuprl Lemma : non-empty-bag-mapfilter-union-of-list
∀[T:Type]
∀P:T ⟶ 𝔹. ∀f:T ⟶ Top. ∀L:bag(T) List. (0 < #(bag-mapfilter(f;λx.P[x];bag-union(L))) ⇐⇒ (∃b∈L. 0 < #([x∈b|P[x]])))
Proof
Definitions occuring in Statement :
bag-union: bag-union(bbs),
bag-size: #(bs),
bag-mapfilter: bag-mapfilter(f;P;bs),
bag-filter: [x∈b|p[x]],
bag: bag(T),
l_exists: (∃x∈L. P[x]),
list: T List,
bool: 𝔹,
less_than: a < b,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
bag-union: bag-union(bbs),
bag-mapfilter: bag-mapfilter(f;P;bs),
bag-size: #(bs),
bag-filter: [x∈b|p[x]],
bag-map: bag-map(f;bs),
member: t ∈ T,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
concat: concat(ll),
iff: P ⇐⇒ Q,
and: P ∧ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
rev_implies: P ⇐ Q,
l_exists: (∃x∈L. P[x]),
exists: ∃x:A. B[x],
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
ge: i ≥ j ,
nat_plus: ℕ+
Lemmas referenced :
length-map,
top_wf,
bag-filter_wf,
bag-union_wf,
list-subtype-bag,
bag_wf,
subtype_rel_self,
list_wf,
bool_wf,
list_induction,
iff_wf,
less_than_wf,
bag-size_wf,
assert_wf,
reduce_nil_lemma,
filter_nil_lemma,
length_of_nil_lemma,
stuck-spread,
base_wf,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
l_exists_wf_nil,
reduce_cons_lemma,
filter_append_sq,
length-append,
decidable__lt,
nat_wf,
l_exists_wf,
l_member_wf,
add-is-int-iff,
intformnot_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
false_wf,
nat_properties,
add_nat_plus,
nat_plus_wf,
nat_plus_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf,
or_wf,
l_exists_cons,
cons_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
cumulativity,
hypothesisEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
because_Cache,
independent_isectElimination,
functionEquality,
universeEquality,
setEquality,
independent_functionElimination,
rename,
dependent_functionElimination,
independent_pairFormation,
imageElimination,
productElimination,
natural_numberEquality,
baseClosed,
setElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
unionElimination,
inlFormation,
inrFormation,
equalityTransitivity,
equalitySymmetry,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
addEquality,
applyLambdaEquality,
dependent_set_memberEquality,
addLevel,
impliesFunctionality
Latex:
\mforall{}[T:Type]
\mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}f:T {}\mrightarrow{} Top. \mforall{}L:bag(T) List.
(0 < \#(bag-mapfilter(f;\mlambda{}x.P[x];bag-union(L))) \mLeftarrow{}{}\mRightarrow{} (\mexists{}b\mmember{}L. 0 < \#([x\mmember{}b|P[x]])))
Date html generated:
2017_10_01-AM-08_47_09
Last ObjectModification:
2017_07_26-PM-04_31_46
Theory : bags
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