Nuprl Lemma : filter_is_sublist
∀[T:Type]. ∀L:T List. ∀P:T ⟶ 𝔹. filter(P;L) ⊆ L
Proof
Definitions occuring in Statement :
sublist: L1 ⊆ L2,
filter: filter(P;l),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀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,
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,
or: P ∨ Q,
cand: A c∧ B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
not: ¬A,
false: False
Lemmas referenced :
list_induction,
all_wf,
bool_wf,
sublist_wf,
filter_wf5,
subtype_rel_dep_function,
l_member_wf,
subtype_rel_self,
set_wf,
list_wf,
filter_nil_lemma,
nil-sublist,
filter_cons_lemma,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
equal_wf,
cons_wf,
cons_sublist_cons,
sublist_tl,
null_cons_lemma,
false_wf,
reduce_tl_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
hypothesis,
applyEquality,
because_Cache,
setEquality,
independent_isectElimination,
setElimination,
rename,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
universeEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
unionElimination,
equalityElimination,
productElimination,
inlFormation,
independent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. filter(P;L) \msubseteq{} L
Date html generated:
2019_06_20-PM-01_24_27
Last ObjectModification:
2018_09_17-PM-05_50_56
Theory : list_1
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