Nuprl Lemma : mapfilter-contains
∀[T,S:Type]. ∀as,bs:T List. ∀P:T ⟶ 𝔹. ∀f:{x:T| ↑(P x)} ⟶ S. (as ⊆ bs ⇒ mapfilter(f;P;as) ⊆ mapfilter(f;P;bs))
Proof
Definitions occuring in Statement :
l_contains: A ⊆ B,
mapfilter: mapfilter(f;P;L),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
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],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
cand: A c∧ B,
exists: ∃x:A. B[x],
guard: {T}
Lemmas referenced :
l_all_iff,
l_member_wf,
mapfilter_wf,
assert_wf,
member-mapfilter,
subtype_rel_dep_function,
bool_wf,
subtype_rel_self,
set_wf,
equal_wf,
l_all_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
lambdaEquality,
cumulativity,
setElimination,
rename,
hypothesis,
setEquality,
productElimination,
independent_functionElimination,
functionExtensionality,
applyEquality,
independent_isectElimination,
because_Cache,
dependent_set_memberEquality,
productEquality,
dependent_pairFormation,
independent_pairFormation,
functionEquality,
universeEquality
Latex:
\mforall{}[T,S:Type].
\mforall{}as,bs:T List. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}f:\{x:T| \muparrow{}(P x)\} {}\mrightarrow{} S.
(as \msubseteq{} bs {}\mRightarrow{} mapfilter(f;P;as) \msubseteq{} mapfilter(f;P;bs))
Date html generated:
2017_04_17-AM-07_30_14
Last ObjectModification:
2017_02_27-PM-04_07_24
Theory : list_1
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