Nuprl Lemma : member_append
∀[T:Type]. ∀x:T. ∀l1,l2:T List.  ((x ∈ l1 @ l2) 
⇐⇒ (x ∈ l1) ∨ (x ∈ l2))
Proof
Definitions occuring in Statement : 
l_member: (x ∈ l)
, 
append: as @ bs
, 
list: T List
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
or: 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
, 
append: as @ bs
, 
so_lambda: so_lambda(x,y,z.t[x; y; z])
, 
top: Top
, 
so_apply: x[s1;s2;s3]
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
guard: {T}
, 
or: P ∨ Q
, 
prop: ℙ
, 
rev_implies: P 
⇐ Q
, 
false: False
Lemmas referenced : 
list_induction, 
all_wf, 
list_wf, 
iff_wf, 
l_member_wf, 
append_wf, 
or_wf, 
list_ind_nil_lemma, 
false_wf, 
nil_member, 
nil_wf, 
list_ind_cons_lemma, 
equal_wf, 
cons_member, 
cons_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
lambdaFormation, 
cut, 
thin, 
lemma_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
because_Cache, 
sqequalRule, 
lambdaEquality, 
hypothesisEquality, 
hypothesis, 
independent_functionElimination, 
dependent_functionElimination, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
independent_pairFormation, 
inrFormation, 
unionElimination, 
addLevel, 
allFunctionality, 
productElimination, 
impliesFunctionality, 
orFunctionality, 
rename, 
inlFormation, 
universeEquality
Latex:
\mforall{}[T:Type].  \mforall{}x:T.  \mforall{}l1,l2:T  List.    ((x  \mmember{}  l1  @  l2)  \mLeftarrow{}{}\mRightarrow{}  (x  \mmember{}  l1)  \mvee{}  (x  \mmember{}  l2))
Date html generated:
2016_05_14-AM-06_42_08
Last ObjectModification:
2015_12_26-PM-00_29_36
Theory : list_0
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