Nuprl Lemma : member-mapfilter-univ
∀[T:Type]
∀L:T List. ∀P:{x:T| (x ∈ L)} ⟶ 𝔹.
∀[T':Type]
∀f:{x:T| (x ∈ L) c∧ (↑(P x))} ⟶ T'. ∀x:T'.
((x ∈ mapfilter(f;P;L)) ⇐⇒ ∃y:T. ((y ∈ L) ∧ ((↑(P y)) c∧ (x = (f y) ∈ T'))))
Proof
Definitions occuring in Statement :
mapfilter: mapfilter(f;P;L),
l_member: (x ∈ l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
cand: A c∧ B,
prop: ℙ,
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
mapfilter: mapfilter(f;P;L),
subtype_rel: A ⊆r B,
uimplies: b supposing a
Lemmas referenced :
istype-universe,
list_wf,
bool_wf,
list-subtype,
assert_wf,
istype-assert,
l_member_wf,
member_map_filter,
l_member-settype,
map_wf,
filter_type,
subtype_rel_list
Rules used in proof :
universeEquality,
instantiate,
inhabitedIsType,
setIsType,
functionIsType,
because_Cache,
universeIsType,
productIsType,
sqequalRule,
independent_pairFormation,
dependent_set_memberEquality_alt,
rename,
setElimination,
applyEquality,
functionExtensionality,
dependent_functionElimination,
hypothesis,
hypothesisEquality,
setEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productElimination,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
equalityIstype,
productEquality,
equalityIsType1,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
cumulativity
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. \mforall{}P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}.
\mforall{}[T':Type]
\mforall{}f:\{x:T| (x \mmember{} L) c\mwedge{} (\muparrow{}(P x))\} {}\mrightarrow{} T'. \mforall{}x:T'.
((x \mmember{} mapfilter(f;P;L)) \mLeftarrow{}{}\mRightarrow{} \mexists{}y:T. ((y \mmember{} L) \mwedge{} ((\muparrow{}(P y)) c\mwedge{} (x = (f y)))))
Date html generated:
2019_10_15-AM-10_21_57
Last ObjectModification:
2019_08_05-PM-02_08_39
Theory : list_1
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