Nuprl Lemma : l_all-mapfilter
∀[T,A:Type].
∀as:T List. ∀p:{a:T| (a ∈ as)} ⟶ 𝔹. ∀f:{a:T| (a ∈ as) ∧ (↑(p a))} ⟶ A. ∀P:A ⟶ ℙ.
((∀x∈mapfilter(f;p;as).P[x]) ⇐⇒ (∀x∈as.(↑(p x)) ⇒ P[f x]))
Proof
Definitions occuring in Statement :
mapfilter: mapfilter(f;P;L),
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
istype: istype(T),
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
exists: ∃x:A. B[x],
guard: {T}
Lemmas referenced :
l_all_wf,
mapfilter_wf,
l_member_wf,
list-subtype,
assert_wf,
subtype_rel_dep_function,
istype-universe,
l_all_iff,
subtype_rel_self,
bool_wf,
list_wf,
member-mapfilter
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setEquality,
hypothesis,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
applyEquality,
setElimination,
rename,
Error :dependent_set_memberEquality_alt,
independent_pairFormation,
sqequalRule,
Error :productIsType,
Error :universeIsType,
instantiate,
cumulativity,
Error :lambdaEquality_alt,
universeEquality,
because_Cache,
Error :setIsType,
independent_isectElimination,
dependent_functionElimination,
functionEquality,
productEquality,
productElimination,
independent_functionElimination,
Error :functionIsType,
Error :inhabitedIsType,
Error :dependent_pairFormation_alt,
Error :equalityIsType1,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[T,A:Type].
\mforall{}as:T List. \mforall{}p:\{a:T| (a \mmember{} as)\} {}\mrightarrow{} \mBbbB{}. \mforall{}f:\{a:T| (a \mmember{} as) \mwedge{} (\muparrow{}(p a))\} {}\mrightarrow{} A. \mforall{}P:A {}\mrightarrow{} \mBbbP{}.
((\mforall{}x\mmember{}mapfilter(f;p;as).P[x]) \mLeftarrow{}{}\mRightarrow{} (\mforall{}x\mmember{}as.(\muparrow{}(p x)) {}\mRightarrow{} P[f x]))
Date html generated:
2019_06_20-PM-01_27_40
Last ObjectModification:
2018_10_05-AM-10_53_17
Theory : list_1
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