Nuprl Lemma : member_tl
∀[T:Type]. ∀as:T List. ∀x:T. (x ∈ tl(as)) ⇒ (x ∈ as) supposing 0 < ||as||
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
length: ||as||,
tl: tl(l),
list: T List,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
squash: ↓T,
prop: ℙ,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
guard: {T},
sq_stable: SqStable(P),
int_seg: {i..j-},
lelt: i ≤ j < k
Lemmas referenced :
member-less_than,
length_wf,
less_than_wf,
squash_wf,
true_wf,
length_tl,
decidable__le,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
zero-add,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel2,
iff_weakening_equal,
not-le-2,
sq_stable__le,
le-add-cancel,
le_wf,
decidable__lt,
not-lt-2,
subtract_wf,
equal_wf,
select_tl,
lelt_wf,
select_wf,
l_member_wf,
tl_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
cumulativity,
hypothesisEquality,
hypothesis,
independent_isectElimination,
rename,
productElimination,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
intEquality,
setElimination,
because_Cache,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
independent_functionElimination,
addEquality,
sqequalRule,
isect_memberEquality,
voidEquality,
minusEquality,
imageMemberEquality,
baseClosed,
universeEquality,
dependent_pairFormation,
dependent_set_memberEquality,
productEquality
Latex:
\mforall{}[T:Type]. \mforall{}as:T List. \mforall{}x:T. (x \mmember{} tl(as)) {}\mRightarrow{} (x \mmember{} as) supposing 0 < ||as||
Date html generated:
2017_04_14-AM-08_39_22
Last ObjectModification:
2017_02_27-PM-03_29_48
Theory : list_0
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