Nuprl Lemma : filter-contains
∀[T:Type]. ∀as:T List. ∀P:T ⟶ 𝔹. filter(P;as) ⊆ as
Proof
Definitions occuring in Statement :
l_contains: A ⊆ B,
filter: filter(P;l),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
l_contains: A ⊆ B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
l_all_iff,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
member_filter_2,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
applyEquality,
because_Cache,
lambdaEquality,
hypothesis,
setEquality,
independent_isectElimination,
setElimination,
rename,
productElimination,
independent_functionElimination,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}as:T List. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. filter(P;as) \msubseteq{} as
Date html generated:
2016_05_14-PM-01_28_58
Last ObjectModification:
2015_12_26-PM-05_22_11
Theory : list_1
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