Nuprl Lemma : member-union
∀[T:Type]. ∀eq:EqDecider(T). ∀as,bs:T List. ∀x:T. ((x ∈ as ⋃ bs) ⇐⇒ (x ∈ as) ∨ (x ∈ bs))
Proof
Definitions occuring in Statement :
l-union: as ⋃ bs,
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
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],
l-union: as ⋃ bs,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
or: P ∨ Q,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
not: ¬A,
false: False
Lemmas referenced :
list_induction,
iff_wf,
l_member_wf,
reduce_wf,
list_wf,
insert_wf,
or_wf,
reduce_nil_lemma,
reduce_cons_lemma,
deq_wf,
nil_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
equal_wf,
member-insert,
cons_member,
cons_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
rename,
because_Cache,
universeEquality,
independent_pairFormation,
inlFormation,
unionElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
inrFormation,
addLevel,
impliesFunctionality,
orFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}eq:EqDecider(T). \mforall{}as,bs:T List. \mforall{}x:T. ((x \mmember{} as \mcup{} bs) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} as) \mvee{} (x \mmember{} bs))
Date html generated:
2016_05_14-PM-03_24_34
Last ObjectModification:
2015_12_26-PM-06_21_54
Theory : decidable!equality
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