Nuprl Lemma : sub-bag-union-of-list
∀[T:Type]. ∀[x:bag(T)]. ∀bs:bag(T) List. ((x ∈ bs) ⇒ sub-bag(T;x;bag-union(bs)))
Proof
Definitions occuring in Statement :
sub-bag: sub-bag(T;as;bs),
bag-union: bag-union(bbs),
bag: bag(T),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
sub-bag: sub-bag(T;as;bs),
member: t ∈ T,
nat: ℕ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
single-bag: {x},
bag-append: as + bs,
true: True,
top: Top,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
bag-append_wf,
bag-union_wf,
firstn_wf,
bag_wf,
list-subtype-bag,
subtype_rel_self,
nth_tl_wf,
equal_wf,
l_member_wf,
list_wf,
firstn_nth_tl_decomp,
lelt_wf,
length_wf,
single-bag_wf,
bag-subtype-list,
add-commutes,
bag-append-assoc-comm,
iff_weakening_equal,
squash_wf,
true_wf,
bag-union-append,
bag-union-single
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
cut,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
setElimination,
rename,
applyEquality,
because_Cache,
independent_isectElimination,
sqequalRule,
addEquality,
natural_numberEquality,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
equalityTransitivity,
universeEquality,
dependent_set_memberEquality,
independent_pairFormation,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaEquality,
imageElimination,
equalityUniverse,
levelHypothesis,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[x:bag(T)]. \mforall{}bs:bag(T) List. ((x \mmember{} bs) {}\mRightarrow{} sub-bag(T;x;bag-union(bs)))
Date html generated:
2017_10_01-AM-08_54_54
Last ObjectModification:
2017_07_26-PM-04_36_45
Theory : bags
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