Nuprl Lemma : bag-append-no-repeats
∀[T:Type]. ∀[as,bs:bag(T)].
uiff(bag-no-repeats(T;as) ∧ bag-no-repeats(T;bs) ∧ (∀z:T. (z ↓∈ as ⇒ (¬z ↓∈ bs)));bag-no-repeats(T;as + 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],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
false: False,
cand: A c∧ B,
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
bag-append: as + bs,
bag: bag(T),
quotient: x,y:A//B[x; y],
iff: P ⇐⇒ Q,
bag-member: x ↓∈ bs,
guard: {T},
l_disjoint: l_disjoint(T;l1;l2)
Lemmas referenced :
bag-no-repeats_wf,
all_wf,
bag-member_wf,
not_wf,
bag-append_wf,
bag_wf,
bag-append-no-repeats1,
bag_to_squash_list,
sq_stable__bag-no-repeats,
list-subtype-bag,
no_repeats_functionality_wrt_permutation,
no_repeats_append_iff,
equal_wf,
no_repeats_wf,
member_wf,
list_wf,
append_wf,
permutation_wf,
member-permutation
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
imageElimination,
hypothesis,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
productEquality,
extract_by_obid,
isectElimination,
cumulativity,
lambdaEquality,
functionEquality,
productElimination,
independent_pairEquality,
dependent_functionElimination,
voidElimination,
because_Cache,
universeEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
independent_functionElimination,
promote_hyp,
hyp_replacement,
applyLambdaEquality,
rename,
applyEquality,
lambdaFormation,
pertypeElimination,
dependent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}[as,bs:bag(T)].
uiff(bag-no-repeats(T;as)
\mwedge{} bag-no-repeats(T;bs)
\mwedge{} (\mforall{}z:T. (z \mdownarrow{}\mmember{} as {}\mRightarrow{} (\mneg{}z \mdownarrow{}\mmember{} bs)));bag-no-repeats(T;as + bs))
Date html generated:
2017_10_01-AM-08_59_13
Last ObjectModification:
2017_07_26-PM-04_41_13
Theory : bags
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