Nuprl Lemma : null-bag-filter-map
∀[T,A:Type]. ∀[f:A ⟶ T]. ∀[p:T ⟶ 𝔹]. ∀[as:bag(A)]. null([x∈bag-map(f;as)|p[x]]) = null([x∈as|p[f x]])
Proof
Definitions occuring in Statement :
bag-filter: [x∈b|p[x]],
bag-map: bag-map(f;bs),
bag: bag(T),
null: null(as),
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
bag-filter: [x∈b|p[x]],
bag-map: bag-map(f;bs),
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_apply: x[s],
compose: f o g,
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
prop: ℙ,
bag-null: bag-null(bs),
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B
Lemmas referenced :
bool_wf,
list_wf,
filter_map,
null-map,
quotient-member-eq,
permutation_wf,
permutation-equiv,
null_wf,
filter_wf5,
l_member_wf,
equal_wf,
equal-wf-base,
bag_wf,
bag-null_wf,
assert_wf,
bag-filter_wf,
list-subtype-bag
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
thin,
equalityTransitivity,
equalitySymmetry,
isectElimination,
cumulativity,
hypothesisEquality,
lambdaFormation,
because_Cache,
rename,
lambdaEquality,
applyEquality,
functionExtensionality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
setElimination,
setEquality,
hyp_replacement,
applyLambdaEquality,
productEquality,
axiomEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T,A:Type]. \mforall{}[f:A {}\mrightarrow{} T]. \mforall{}[p:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[as:bag(A)].
null([x\mmember{}bag-map(f;as)|p[x]]) = null([x\mmember{}as|p[f x]])
Date html generated:
2017_10_01-AM-08_45_43
Last ObjectModification:
2017_07_26-PM-04_30_53
Theory : bags
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