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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