Nuprl Lemma : member-merge
∀[T:Type]. ∀bs,as:T List. ∀x:T. ((x ∈ merge(as;bs)) ⇐⇒ (x ∈ as) ∨ (x ∈ bs)) supposing T ⊆r ℤ
Proof
Definitions occuring in Statement :
merge: merge(as;bs),
l_member: (x ∈ l),
list: T List,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
merge: merge(as;bs),
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
or: P ∨ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
false: False,
guard: {T}
Lemmas referenced :
list_induction,
all_wf,
list_wf,
iff_wf,
l_member_wf,
merge_wf,
or_wf,
reduce_nil_lemma,
false_wf,
nil_member,
nil_wf,
reduce_cons_lemma,
equal_wf,
cons_member,
cons_wf,
member-s-insert,
s-insert_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
axiomEquality,
hypothesis,
thin,
rename,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
lambdaEquality,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
inlFormation,
unionElimination,
because_Cache,
addLevel,
allFunctionality,
productElimination,
impliesFunctionality,
orFunctionality,
inrFormation,
cumulativity,
intEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}bs,as:T List. \mforall{}x:T. ((x \mmember{} merge(as;bs)) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} as) \mvee{} (x \mmember{} bs)) supposing T \msubseteq{}r \mBbbZ{}
Date html generated:
2016_05_15-PM-03_52_31
Last ObjectModification:
2015_12_27-PM-01_23_47
Theory : general
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