Nuprl Lemma : bag-filter_wf
∀[T:Type]. ∀[p:T ⟶ 𝔹]. ∀[bs:bag(T)]. ([x∈bs|p[x]] ∈ bag({x:T| ↑p[x]} ))
Proof
Definitions occuring in Statement :
bag-filter: [x∈b|p[x]],
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
bag: bag(T),
so_apply: x[s],
prop: ℙ,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bag-filter: [x∈b|p[x]],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
so_lambda: λ2x.t[x]
Lemmas referenced :
bag_wf,
assert_wf,
list_wf,
quotient-member-eq,
filter_type,
permutation_wf,
equal_wf,
equal-wf-base,
bool_wf,
permutation-filter,
permutation-equiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
setEquality,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
because_Cache,
rename,
independent_isectElimination,
dependent_functionElimination,
lambdaEquality,
independent_functionElimination,
productEquality,
axiomEquality,
isect_memberEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[p:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[bs:bag(T)]. ([x\mmember{}bs|p[x]] \mmember{} bag(\{x:T| \muparrow{}p[x]\} ))
Date html generated:
2017_10_01-AM-08_45_17
Last ObjectModification:
2017_07_26-PM-04_30_38
Theory : bags
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