Nuprl Lemma : sub-bag-mapfilter-implies
∀[A,B:Type].
∀as:bag(A). ∀bs:bag(B). ∀f:A ⟶ B. ∀P:A ⟶ 𝔹. (sub-bag(B;bag-map(f;as);bs) ⇒ sub-bag(B;bag-mapfilter(f;P;as);bs))
Proof
Definitions occuring in Statement :
sub-bag: sub-bag(T;as;bs),
bag-mapfilter: bag-mapfilter(f;P;bs),
bag-map: bag-map(f;bs),
bag: bag(T),
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
sub-bag: sub-bag(T;as;bs),
bag-mapfilter: bag-mapfilter(f;P;bs),
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top
Lemmas referenced :
bag-filter-split,
equal_wf,
bag_wf,
bag-append_wf,
bag-map_wf,
bag-map-append,
bag-filter_wf,
subtype_rel_bag,
top_wf,
assert_wf,
bnot_wf,
subtype_rel_dep_function,
set_wf,
bag-append-assoc,
sub-bag_wf,
bool_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalitySymmetry,
hypothesis,
hyp_replacement,
Error :applyLambdaEquality,
cumulativity,
equalityTransitivity,
functionExtensionality,
applyEquality,
sqequalRule,
lambdaEquality,
setEquality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
dependent_pairFormation,
setElimination,
rename,
functionEquality
Latex:
\mforall{}[A,B:Type].
\mforall{}as:bag(A). \mforall{}bs:bag(B). \mforall{}f:A {}\mrightarrow{} B. \mforall{}P:A {}\mrightarrow{} \mBbbB{}.
(sub-bag(B;bag-map(f;as);bs) {}\mRightarrow{} sub-bag(B;bag-mapfilter(f;P;as);bs))
Date html generated:
2016_10_25-AM-10_38_00
Last ObjectModification:
2016_07_12-AM-06_51_32
Theory : bags
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