Nuprl Lemma : member-firstn-implies-member
∀[T:Type]. ∀x:T. ∀L:T List. ∀n:ℤ. ((x ∈ firstn(n;L)) ⇒ (x ∈ L))
Proof
Definitions occuring in Statement :
firstn: firstn(n;as),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
firstn: firstn(n;as),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
uimplies: b supposing a,
not: ¬A,
false: False,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff
Lemmas referenced :
list_induction,
all_wf,
l_member_wf,
firstn_wf,
list_wf,
list_ind_nil_lemma,
nil_wf,
list_ind_cons_lemma,
ifthenelse_wf,
lt_int_wf,
cons_wf,
subtract_wf,
assert_wf,
bnot_wf,
not_wf,
less_than_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
cons_member,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
intEquality,
functionEquality,
hypothesisEquality,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
rename,
natural_numberEquality,
universeEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
unionElimination,
instantiate,
cumulativity,
productElimination,
independent_pairFormation,
impliesFunctionality,
inlFormation,
inrFormation
Latex:
\mforall{}[T:Type]. \mforall{}x:T. \mforall{}L:T List. \mforall{}n:\mBbbZ{}. ((x \mmember{} firstn(n;L)) {}\mRightarrow{} (x \mmember{} L))
Date html generated:
2016_05_14-PM-01_27_15
Last ObjectModification:
2015_12_26-PM-04_50_54
Theory : list_1
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