Nuprl Lemma : assert-deq-member
∀[A:Type]. ∀eq:EqDecider(A). ∀L:A List. ∀x:A. (↑x ∈b L ⇐⇒ (x ∈ L))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
deq-member: x ∈b L,
list: T List,
deq: EqDecider(T),
assert: ↑b,
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],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
false: False,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
not: ¬A,
deq-member: x ∈b L,
or: P ∨ Q,
guard: {T},
uiff: uiff(P;Q),
eqof: eqof(d)
Lemmas referenced :
list_induction,
all_wf,
iff_wf,
assert_wf,
deq-member_wf,
l_member_wf,
list_wf,
deq_member_nil_lemma,
deq_member_cons_lemma,
deq_wf,
false_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
equal_wf,
or_wf,
cons_member,
cons_wf,
bor_wf,
eqof_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
safe-assert-deq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
rename,
because_Cache,
universeEquality,
independent_pairFormation,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
unionElimination,
inlFormation,
inrFormation,
addLevel,
productElimination,
impliesFunctionality,
applyEquality,
orFunctionality,
levelHypothesis,
andLevelFunctionality,
impliesLevelFunctionality
Latex:
\mforall{}[A:Type]. \mforall{}eq:EqDecider(A). \mforall{}L:A List. \mforall{}x:A. (\muparrow{}x \mmember{}\msubb{} L \mLeftarrow{}{}\mRightarrow{} (x \mmember{} L))
Date html generated:
2017_04_14-AM-08_53_31
Last ObjectModification:
2017_02_27-PM-03_38_03
Theory : list_0
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