Nuprl Lemma : sublist*_filter
∀[T:Type]. ∀P:T ⟶ 𝔹. ∀as,bs:T List. (sublist*(T;as;bs) ⇒ sublist*(T;filter(P;as);filter(P;bs)))
Proof
Definitions occuring in Statement :
sublist*: sublist*(T;as;bs),
filter: filter(P;l),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
sublist*: sublist*(T;as;bs),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
l_subset: l_subset(T;as;bs)
Lemmas referenced :
l_subset_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
sublist_wf,
list_wf,
all_wf,
sublist_filter,
member_filter
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
lambdaEquality,
hypothesis,
setEquality,
independent_isectElimination,
setElimination,
rename,
because_Cache,
functionEquality,
universeEquality,
dependent_functionElimination,
productElimination,
independent_functionElimination,
independent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}as,bs:T List. (sublist*(T;as;bs) {}\mRightarrow{} sublist*(T;filter(P;as);filter(P;bs)))
Date html generated:
2019_10_15-AM-10_58_38
Last ObjectModification:
2018_09_17-PM-06_29_24
Theory : list!
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