Nuprl Lemma : set-equal-nil
∀[T:Type]. ∀bs:T List. (set-equal(T;[];bs) ⇐⇒ ↑null(bs))
Proof
Definitions occuring in Statement :
set-equal: set-equal(T;x;y),
null: null(as),
nil: [],
list: T List,
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],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
or: P ∨ Q,
cons: [a / b],
prop: ℙ,
rev_implies: P ⇐ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
top: Top,
bfalse: ff,
false: False,
set-equal: set-equal(T;x;y)
Lemmas referenced :
assert_of_null,
list-cases,
nil_wf,
product_subtype_list,
assert_witness,
null_wf,
set-equal_wf,
null_nil_lemma,
set-equal-reflex,
null_cons_lemma,
assert_wf,
list_wf,
false_wf,
or_wf,
equal_wf,
l_member_wf,
member_wf,
nil_member,
cons_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
promote_hyp,
hypothesis_subsumption,
sqequalRule,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
universeEquality,
because_Cache,
inlFormation,
addLevel,
impliesFunctionality,
levelHypothesis,
andLevelFunctionality,
impliesLevelFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}bs:T List. (set-equal(T;[];bs) \mLeftarrow{}{}\mRightarrow{} \muparrow{}null(bs))
Date html generated:
2016_05_14-PM-01_38_18
Last ObjectModification:
2015_12_26-PM-05_28_15
Theory : list_1
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