Nuprl Lemma : bag-filter-wf2
∀[T:Type]. ∀[bs:bag(T)]. ∀[p:{b:T| b ↓∈ bs} ⟶ 𝔹]. ([x∈bs|p[x]] ∈ bag({x:{b:T| b ↓∈ bs} | ↑p[x]} ))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
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,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x]
Lemmas referenced :
bag-filter_wf,
bag-member_wf,
bag-subtype,
bool_wf,
bag_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
cumulativity,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
functionEquality,
isect_memberEquality,
because_Cache,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[bs:bag(T)]. \mforall{}[p:\{b:T| b \mdownarrow{}\mmember{} bs\} {}\mrightarrow{} \mBbbB{}]. ([x\mmember{}bs|p[x]] \mmember{} bag(\{x:\{b:T| b \mdownarrow{}\mmember{} bs\} | \muparrow{}p[x]\} )\000C)
Date html generated:
2016_05_15-PM-02_47_12
Last ObjectModification:
2015_12_27-AM-09_36_19
Theory : bags
Home
Index