Nuprl Lemma : remove-repeats-filter
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[P:T ⟶ 𝔹]. ∀[L:T List].
(remove-repeats(eq;filter(P;L)) = filter(P;remove-repeats(eq;L)) ∈ (T List))
Proof
Definitions occuring in Statement :
remove-repeats: remove-repeats(eq;L),
filter: filter(P;l),
list: T List,
deq: EqDecider(T),
bool: 𝔹,
uall: ∀[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,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
all: ∀x:A. B[x],
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 ,
squash: ↓T,
true: True,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
deq: EqDecider(T),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
eqof: eqof(d),
band: p ∧b q
Lemmas referenced :
list_induction,
equal_wf,
list_wf,
remove-repeats_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
set_wf,
subtype_rel_self,
filter_nil_lemma,
remove_repeats_nil_lemma,
nil_wf,
filter_cons_lemma,
remove_repeats_cons_lemma,
eqtt_to_assert,
cons_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
filter-filter,
band_wf,
bnot_wf,
iff_weakening_equal,
iff_imp_equal_bool,
assert_elim,
and_wf,
not_assert_elim,
btrue_neq_bfalse,
assert_wf,
not_wf,
iff_transitivity,
eqof_wf,
iff_weakening_uiff,
assert_of_band,
assert_of_bnot,
safe-assert-deq,
iff_wf,
deq_wf,
band_commutes
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
because_Cache,
applyEquality,
setEquality,
independent_isectElimination,
setElimination,
rename,
lambdaFormation,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
functionExtensionality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
promote_hyp,
instantiate,
independent_pairFormation,
addLevel,
levelHypothesis,
dependent_set_memberEquality,
applyLambdaEquality,
productEquality,
impliesFunctionality,
andLevelFunctionality,
impliesLevelFunctionality,
axiomEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List].
(remove-repeats(eq;filter(P;L)) = filter(P;remove-repeats(eq;L)))
Date html generated:
2017_04_17-AM-09_10_35
Last ObjectModification:
2017_02_27-PM-05_19_19
Theory : decidable!equality
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