Nuprl Lemma : last-mapfilter2
∀[A,B:Type]. ∀[f:A ⟶ B]. ∀[P:A ⟶ 𝔹]. ∀[L:A List].
(last(mapfilter(f;P;L)) = (f last(L)) ∈ B) supposing ((↑(P last(L))) and (¬↑null(mapfilter(f;P;L))))
Proof
Definitions occuring in Statement :
mapfilter: mapfilter(f;P;L),
last: last(L),
null: null(as),
list: T List,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
null: null(as),
mapfilter: mapfilter(f;P;L),
map: map(f;as),
list_ind: list_ind,
filter: filter(P;l),
reduce: reduce(f;k;as),
nil: [],
it: ⋅,
btrue: tt,
true: True,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
false: False,
top: Top,
bool: 𝔹,
unit: Unit,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
assert_of_null,
assert_wf,
null_wf,
mapfilter_wf,
subtype_rel_dep_function,
set_wf,
last-mapfilter,
filter_wf5,
bool_wf,
l_member_wf,
subtype_rel_self,
eqtt_to_assert,
map_nil_lemma,
null_nil_lemma,
map_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
equal-wf-T-base,
list_wf,
last_wf,
not_wf,
list_induction,
null-mapfilter,
filter_nil_lemma,
mapfilter_nil_lemma,
nil_wf,
equal-wf-base,
filter_cons_lemma,
null_cons_lemma,
subtype_rel_list,
top_wf,
cons_wf,
false_wf,
null-map,
last-cons,
length_wf_nat,
nat_wf,
squash_wf,
true_wf,
iff_weakening_equal,
null-filter2,
l_all_iff,
last_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
lambdaFormation,
introduction,
sqequalHypSubstitution,
independent_functionElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
natural_numberEquality,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
cumulativity,
functionExtensionality,
applyEquality,
sqequalRule,
lambdaEquality,
setEquality,
setElimination,
rename,
because_Cache,
voidElimination,
promote_hyp,
isect_memberEquality,
voidEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
dependent_functionElimination,
dependent_pairFormation,
instantiate,
baseClosed,
functionEquality,
universeEquality,
addLevel,
impliesFunctionality,
levelHypothesis,
dependent_set_memberEquality,
imageElimination,
imageMemberEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} B]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:A List].
(last(mapfilter(f;P;L)) = (f last(L))) supposing ((\muparrow{}(P last(L))) and (\mneg{}\muparrow{}null(mapfilter(f;P;L))))
Date html generated:
2017_04_17-AM-07_52_44
Last ObjectModification:
2017_02_27-PM-04_26_42
Theory : list_1
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