Nuprl Lemma : member-nth-tl-implies-member
∀[T:Type]. ∀x:T. ∀n:ℕ. ∀L:T List. ((x ∈ nth_tl(n;L)) ⇒ (x ∈ L))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
nth_tl: nth_tl(n;as),
list: T List,
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
nth_tl: nth_tl(n;as),
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
subtract: n - m,
btrue: tt,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
cons: [a / b],
top: Top
Lemmas referenced :
l_member_wf,
list_wf,
ifthenelse_wf,
le_int_wf,
nth_tl_wf,
tl_wf,
subtract_wf,
all_wf,
set_wf,
less_than_wf,
primrec-wf2,
nat_wf,
assert_wf,
bnot_wf,
not_wf,
le_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
list-cases,
reduce_tl_nil_lemma,
nth_tl_nil,
product_subtype_list,
reduce_tl_cons_lemma,
cons_member,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
sqequalRule,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
rename,
setElimination,
natural_numberEquality,
because_Cache,
lambdaEquality,
functionEquality,
intEquality,
introduction,
universeEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
unionElimination,
instantiate,
cumulativity,
independent_isectElimination,
independent_functionElimination,
productElimination,
independent_pairFormation,
impliesFunctionality,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidElimination,
voidEquality,
inrFormation
Latex:
\mforall{}[T:Type]. \mforall{}x:T. \mforall{}n:\mBbbN{}. \mforall{}L:T List. ((x \mmember{} nth\_tl(n;L)) {}\mRightarrow{} (x \mmember{} L))
Date html generated:
2016_05_14-PM-01_27_21
Last ObjectModification:
2015_12_26-PM-04_51_02
Theory : list_1
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