Nuprl Lemma : member-exists
∀[T:Type]. ∀L:T List. (∃x:T. (x ∈ L) ⇐⇒ ¬(L = [] ∈ (T List)))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
nil: [],
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
not: ¬A,
false: False,
member: t ∈ T,
or: P ∨ Q,
exists: ∃x:A. B[x],
cons: [a / b],
top: Top,
guard: {T},
nat: ℕ,
le: A ≤ B,
decidable: Dec(P),
rev_implies: P ⇐ Q,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
listp: A List+,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
listp-not-nil,
list-cases,
length_of_nil_lemma,
nil_member,
product_subtype_list,
length_of_cons_lemma,
length_wf_nat,
nat_wf,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
equal_wf,
less_than_wf,
length_wf,
equal-wf-T-base,
list_wf,
exists_wf,
l_member_wf,
member_exists,
nil_wf,
cons_wf,
cons_neq_nil,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
unionElimination,
sqequalRule,
productElimination,
independent_functionElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
setElimination,
rename,
natural_numberEquality,
addEquality,
independent_isectElimination,
applyEquality,
lambdaEquality,
intEquality,
because_Cache,
minusEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
cumulativity,
baseClosed,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. (\mexists{}x:T. (x \mmember{} L) \mLeftarrow{}{}\mRightarrow{} \mneg{}(L = []))
Date html generated:
2017_04_17-AM-07_30_55
Last ObjectModification:
2017_02_27-PM-04_08_15
Theory : list_1
Home
Index