Nuprl Lemma : member-reverse
∀[T:Type]. ∀L:T List. ∀x:T. ((x ∈ rev(L)) ⇐⇒ (x ∈ L))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
reverse: rev(as),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
squash: ↓T,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
subtract: n - m,
uiff: uiff(P;Q),
nat_plus: ℕ+,
less_than: a < b,
less_than': less_than'(a;b),
not: ¬A,
false: False,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T)
Lemmas referenced :
l_member_wf,
reverse_wf,
list_wf,
equal_wf,
squash_wf,
true_wf,
select-reverse,
lelt_wf,
length_wf,
iff_weakening_equal,
length-reverse,
subtract_wf,
non_neg_length,
length_wf_nat,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
less_than_wf,
select_wf,
sq_stable__le,
add-associates,
minus-one-mul,
add-swap,
add-commutes,
less-iff-le,
add_functionality_wrt_le,
le_reflexive,
minus-one-mul-top,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
add-zero,
not-le-2,
minus-zero,
zero-add,
omega-shadow,
mul-distributes,
mul-associates,
le-add-cancel,
not-lt-2,
minus-add,
minus-minus,
nat_properties,
decidable__le,
decidable__lt,
subtype_base_sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
universeEquality,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
setElimination,
rename,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_pairFormation,
sqequalIntensionalEquality,
intEquality,
dependent_functionElimination,
promote_hyp,
productEquality,
addLevel,
addEquality,
multiplyEquality,
levelHypothesis,
minusEquality,
unionElimination,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}x:T. ((x \mmember{} rev(L)) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} L))
Date html generated:
2017_04_14-AM-08_40_40
Last ObjectModification:
2017_02_27-PM-03_31_13
Theory : list_0
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