Nuprl Lemma : deq-member-union
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[as,bs:T List]. ∀[x:T]. (x ∈b as ⋃ bs ~ x ∈b as ∨bx ∈b bs)
Proof
Definitions occuring in Statement :
l-union: as ⋃ bs,
deq-member: x ∈b L,
list: T List,
deq: EqDecider(T),
bor: p ∨bq,
uall: ∀[x:A]. B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
prop: ℙ,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q
Lemmas referenced :
subtype_base_sq,
bool_subtype_base,
iff_imp_equal_bool,
deq-member_wf,
l-union_wf,
bor_wf,
list_wf,
deq_wf,
assert_wf,
or_wf,
l_member_wf,
iff_wf,
assert-deq-member,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
member-union
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
independent_isectElimination,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
sqequalAxiom,
sqequalRule,
isect_memberEquality,
universeEquality,
addLevel,
productElimination,
independent_pairFormation,
impliesFunctionality,
lambdaFormation,
orFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[as,bs:T List]. \mforall{}[x:T]. (x \mmember{}\msubb{} as \mcup{} bs \msim{} x \mmember{}\msubb{} as \mvee{}\msubb{}x \mmember{}\msubb{} bs)
Date html generated:
2016_05_14-PM-03_25_01
Last ObjectModification:
2015_12_26-PM-06_22_21
Theory : decidable!equality
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