Nuprl Lemma : mapfilter-bor-eq
∀T,U:Type. ∀f:T ─→ U. ∀P,Q:T ─→ 𝔹. ∀L:T List.
((mapfilter(f;P;L) @ mapfilter(f;Q;L))
= (mapfilter(f;λx.(P[x] ∨bQ[x]);L) @ mapfilter(f;λx.(P[x] ∧b Q[x]);L))
∈ bag(U))
Proof
Definitions occuring in Statement :
mapfilter: mapfilter(f;P;L),
append: as @ bs,
list: T List,
bor: p ∨bq,
band: p ∧b q,
bool: 𝔹,
so_apply: x[s],
all: ∀x:A. B[x],
lambda: λx.A[x],
function: x:A ─→ B[x],
universe: Type,
equal: s = t ∈ T,
bag: bag(T)
Lemmas :
list_induction,
equal_wf,
bag_wf,
append_wf,
mapfilter_wf,
subtype_rel_dep_function,
assert_wf,
set_wf,
list-subtype-bag,
bor_wf,
bool_wf,
eqtt_to_assert,
filter_nil_lemma,
map_nil_lemma,
list_ind_nil_lemma,
nil_wf,
list_ind_cons_lemma,
list_wf,
mapfilter-append,
mapfilter-singleton,
map_cons_lemma,
equal-wf-T-base,
or_wf,
bnot_wf,
not_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
eqff_to_assert,
assert-bnot,
cons-bag_wf,
uiff_transitivity,
assert_of_bnot,
assert_of_bor,
iff_transitivity,
iff_weakening_uiff,
bnot_thru_bor,
assert_of_band,
bnot_thru_band,
squash_wf,
true_wf,
iff_weakening_equal,
bag-append-assoc,
bag-append-comm,
cons_wf,
bag-append_wf
Latex:
\mforall{}T,U:Type. \mforall{}f:T {}\mrightarrow{} U. \mforall{}P,Q:T {}\mrightarrow{} \mBbbB{}. \mforall{}L:T List.
((mapfilter(f;P;L) @ mapfilter(f;Q;L))
= (mapfilter(f;\mlambda{}x.(P[x] \mvee{}\msubb{}Q[x]);L) @ mapfilter(f;\mlambda{}x.(P[x] \mwedge{}\msubb{} Q[x]);L)))
Date html generated:
2015_07_21-PM-04_52_39
Last ObjectModification:
2015_02_04-PM-05_09_12
Home
Index