Nuprl Lemma : filter-union
∀[T:Type]. ∀eq:EqDecider(T). ∀as,bs:T List. ∀P:T ⟶ 𝔹. (filter(P;as ⋃ bs) = filter(P;as) ⋃ filter(P;bs) ∈ (T List))
Proof
Definitions occuring in Statement :
l-union: as ⋃ bs,
filter: filter(P;l),
list: T List,
deq: EqDecider(T),
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
l-union: as ⋃ bs,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
implies: P ⇒ Q,
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
squash: ↓T,
not: ¬A,
iff: P ⇐⇒ Q,
true: True,
rev_implies: P ⇐ Q,
cand: A c∧ B
Lemmas referenced :
list_induction,
equal_wf,
list_wf,
filter_wf5,
reduce_wf,
insert_wf,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
reduce_nil_lemma,
filter_nil_lemma,
reduce_cons_lemma,
filter_cons_lemma,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
deq_wf,
insert-cases2,
squash_wf,
true_wf,
ite_rw_false,
cons_wf,
member_filter_2,
assert_wf,
iff_weakening_equal,
member_filter
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
because_Cache,
applyEquality,
setEquality,
independent_isectElimination,
setElimination,
rename,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
functionExtensionality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
functionEquality,
axiomEquality,
universeEquality,
hyp_replacement,
applyLambdaEquality,
imageElimination,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_pairFormation
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}as,bs:T List. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. (filter(P;as \mcup{} bs) = filter(P;as) \mcup{} filter(P;bs))
Date html generated:
2017_04_17-AM-09_09_50
Last ObjectModification:
2017_02_27-PM-05_17_56
Theory : decidable!equality
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