Nuprl Lemma : sub-bag-filter3
∀T:Type. ∀p:T ⟶ 𝔹. ∀b,c:bag(T). (sub-bag(T;b;c) ⇒ sub-bag(T;[x∈b|p[x]];[x∈c|p[x]]))
Proof
Definitions occuring in Statement :
sub-bag: sub-bag(T;as;bs),
bag-filter: [x∈b|p[x]],
bag: bag(T),
bool: 𝔹,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
sub-bag: sub-bag(T;as;bs),
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
top: Top
Lemmas referenced :
bag-filter_wf,
subtype_rel_bag,
assert_wf,
equal_wf,
bag_wf,
bag-append_wf,
sub-bag_wf,
bool_wf,
bag-filter-append
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
cut,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesis,
setEquality,
independent_isectElimination,
setElimination,
rename,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
isect_memberEquality,
voidElimination,
voidEquality,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}T:Type. \mforall{}p:T {}\mrightarrow{} \mBbbB{}. \mforall{}b,c:bag(T). (sub-bag(T;b;c) {}\mRightarrow{} sub-bag(T;[x\mmember{}b|p[x]];[x\mmember{}c|p[x]]))
Date html generated:
2016_10_25-AM-10_30_32
Last ObjectModification:
2016_07_12-AM-06_46_44
Theory : bags
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