Nuprl Lemma : bag-no-repeats-append
∀[T:Type]. ∀[as,bs:bag(T)].
uiff(bag-no-repeats(T;as + bs);bag-no-repeats(T;as) ∧ bag-no-repeats(T;bs) ∧ (∀x:T. (x ↓∈ as ⇒ (¬x ↓∈ bs))))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-no-repeats: bag-no-repeats(T;bs),
bag-append: as + bs,
bag: bag(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
bag-no-repeats: bag-no-repeats(T;bs),
squash: ↓T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
false: False
Lemmas referenced :
bag_wf,
uiff_wf,
bag-append-no-repeats,
bag-member_wf,
not_wf,
rev_implies_wf,
all_wf,
and_wf,
iff_weakening_uiff,
bag-append_wf,
bag-no-repeats_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
introduction,
hypothesis,
sqequalRule,
sqequalHypSubstitution,
imageElimination,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
lemma_by_obid,
isectElimination,
because_Cache,
addLevel,
productElimination,
independent_isectElimination,
lambdaEquality,
independent_functionElimination,
independent_pairEquality,
dependent_functionElimination,
voidElimination,
cumulativity,
functionEquality,
universeEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}[as,bs:bag(T)].
uiff(bag-no-repeats(T;as + bs);bag-no-repeats(T;as)
\mwedge{} bag-no-repeats(T;bs)
\mwedge{} (\mforall{}x:T. (x \mdownarrow{}\mmember{} as {}\mRightarrow{} (\mneg{}x \mdownarrow{}\mmember{} bs))))
Date html generated:
2016_05_15-PM-02_56_04
Last ObjectModification:
2016_01_16-AM-08_39_39
Theory : bags
Home
Index