Nuprl Lemma : last-filter1
∀[A:Type]. ∀[P:A ⟶ 𝔹]. ∀[L:A List].
(last(filter(P;L)) = last(L) ∈ A) supposing ((↑(P last(L))) and (¬↑null(filter(P;L))))
Proof
Definitions occuring in Statement :
last: last(L),
null: null(as),
filter: filter(P;l),
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),
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
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,
bfalse: ff,
bool: 𝔹,
unit: Unit,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb
Lemmas referenced :
assert_of_null,
assert_wf,
null_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
list_induction,
not_wf,
last_wf,
equal_wf,
null_nil_lemma,
filter_nil_lemma,
true_wf,
null_cons_lemma,
filter_cons_lemma,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
list_wf,
subtype_rel_list,
top_wf,
member-implies-null-eq-bfalse,
and_wf,
btrue_neq_bfalse,
equal-wf-T-base,
btrue_wf,
false_wf,
last_cons2,
null-filter2,
last_member,
l_all_fwd
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,
applyEquality,
sqequalRule,
lambdaEquality,
setEquality,
setElimination,
rename,
because_Cache,
voidElimination,
promote_hyp,
functionEquality,
functionExtensionality,
dependent_functionElimination,
isect_memberEquality,
voidEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
dependent_pairFormation,
instantiate,
universeEquality,
dependent_set_memberEquality,
independent_pairFormation,
baseClosed,
addLevel,
impliesFunctionality
Latex:
\mforall{}[A:Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:A List].
(last(filter(P;L)) = last(L)) supposing ((\muparrow{}(P last(L))) and (\mneg{}\muparrow{}null(filter(P;L))))
Date html generated:
2017_04_17-AM-07_52_55
Last ObjectModification:
2017_02_27-PM-04_26_08
Theory : list_1
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