Nuprl Lemma : mapfilter-subbag
∀T,U:Type. ∀f:T ─→ U. ∀P,Q:T ─→ 𝔹. ∀L:T List.
((∀t:T. ((↑(P t)) ⇒ (↑(Q t)))) ⇒ sub-bag(U;mapfilter(f;P;L);mapfilter(f;Q;L)))
Proof
Definitions occuring in Statement :
mapfilter: mapfilter(f;P;L),
list: T List,
assert: ↑b,
bool: 𝔹,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ─→ B[x],
universe: Type,
sub-bag: sub-bag(T;as;bs)
Lemmas :
mapfilter_wf,
bnot_wf,
bool_wf,
eqtt_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
eqff_to_assert,
assert-bnot,
subtype_rel_dep_function,
assert_wf,
set_wf,
list-subtype-bag,
mapfilter-bor-eq,
bag_wf,
bag-append_wf,
all_wf,
list_wf,
append_wf,
bfalse_wf,
iff_imp_equal_bool,
btrue_wf,
true_wf,
not_wf,
false_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
assert_of_bnot,
iff_wf,
mapfilter-nil,
l_member_wf,
int_seg_wf,
length_wf,
append-nil,
subtype_rel_list,
top_wf
Latex:
\mforall{}T,U:Type. \mforall{}f:T {}\mrightarrow{} U. \mforall{}P,Q:T {}\mrightarrow{} \mBbbB{}. \mforall{}L:T List.
((\mforall{}t:T. ((\muparrow{}(P t)) {}\mRightarrow{} (\muparrow{}(Q t)))) {}\mRightarrow{} sub-bag(U;mapfilter(f;P;L);mapfilter(f;Q;L)))
Date html generated:
2015_07_21-PM-04_52_50
Last ObjectModification:
2015_01_28-AM-08_43_47
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