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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