Nuprl Lemma : bag-member-mapfilter
∀[T,U:Type]. ∀[P:T ⟶ 𝔹]. ∀[x:U]. ∀[bs:bag(T)]. ∀[f:{x:T| ↑(P x)} ⟶ U].
uiff(x ↓∈ bag-mapfilter(f;P;bs);↓∃v:{x:T| ↑(P x)} . (v ↓∈ bs ∧ (x = (f v) ∈ U)))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-mapfilter: bag-mapfilter(f;P;bs),
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
bag: bag(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
uimplies: b supposing a,
quotient: x,y:A//B[x; y],
implies: P ⇒ Q,
bag-mapfilter: bag-mapfilter(f;P;bs),
bag-filter: [x∈b|p[x]],
bag-map: bag-map(f;bs),
mapfilter: mapfilter(f;P;L),
bag-member: x ↓∈ bs,
squash: ↓T,
subtype_rel: A ⊆r B,
cand: A c∧ B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
uiff_wf,
bag-member_wf,
bag-mapfilter_wf,
assert_wf,
squash_wf,
exists_wf,
equal_wf,
list_wf,
list-subtype-bag,
permutation_wf,
equal-wf-base,
bag_wf,
bool_wf,
member-permutation,
member_wf,
mapfilter_wf,
member-mapfilter,
l_member_wf,
subtype_rel_dep_function,
subtype_rel_sets,
assert_functionality_wrt_uiff,
eta_conv,
set_wf
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
cut,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
because_Cache,
dependent_functionElimination,
functionExtensionality,
applyEquality,
setEquality,
hypothesis,
sqequalRule,
lambdaEquality,
lambdaFormation,
setElimination,
rename,
productEquality,
dependent_set_memberEquality,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_pairEquality,
isect_memberEquality,
imageElimination,
independent_isectElimination,
independent_functionElimination,
functionEquality,
universeEquality,
isect_memberFormation,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
independent_pairFormation
Latex:
\mforall{}[T,U:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[x:U]. \mforall{}[bs:bag(T)]. \mforall{}[f:\{x:T| \muparrow{}(P x)\} {}\mrightarrow{} U].
uiff(x \mdownarrow{}\mmember{} bag-mapfilter(f;P;bs);\mdownarrow{}\mexists{}v:\{x:T| \muparrow{}(P x)\} . (v \mdownarrow{}\mmember{} bs \mwedge{} (x = (f v))))
Date html generated:
2017_10_01-AM-08_54_24
Last ObjectModification:
2017_07_26-PM-04_36_04
Theory : bags
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