Nuprl Lemma : bag-null-bag-union
∀[T:Type]. ∀[bbs:bag(bag(T))].  ↑bag-null(bag-union(bbs)) supposing ∀bs:bag(T). (bs ↓∈ bbs 
⇒ (↑bag-null(bs)))
Proof
Definitions occuring in Statement : 
bag-member: x ↓∈ bs
, 
bag-union: bag-union(bbs)
, 
bag-null: bag-null(bs)
, 
bag: bag(T)
, 
assert: ↑b
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
squash: ↓T
, 
so_lambda: λ2x.t[x]
, 
implies: P 
⇒ Q
, 
prop: ℙ
, 
all: ∀x:A. B[x]
, 
so_apply: x[s]
, 
sq_stable: SqStable(P)
, 
exists: ∃x:A. B[x]
, 
subtype_rel: A ⊆r B
, 
empty-bag: {}
, 
bag-null: bag-null(bs)
, 
null: null(as)
, 
bag-union: bag-union(bbs)
, 
concat: concat(ll)
, 
reduce: reduce(f;k;as)
, 
list_ind: list_ind, 
nil: []
, 
it: ⋅
, 
btrue: tt
, 
assert: ↑b
, 
ifthenelse: if b then t else f fi 
, 
true: True
, 
cons-bag: x.b
, 
top: Top
, 
guard: {T}
, 
uiff: uiff(P;Q)
, 
and: P ∧ Q
, 
rev_uimplies: rev_uimplies(P;Q)
, 
cand: A c∧ B
, 
sq_or: a ↓∨ b
, 
or: P ∨ Q
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
Lemmas referenced : 
bag_to_squash_list, 
bag_wf, 
sq_stable__all, 
all_wf, 
bag-member_wf, 
assert_wf, 
bag-null_wf, 
bag-union_wf, 
sq_stable_from_decidable, 
decidable__assert, 
assert_witness, 
squash_wf, 
list_induction, 
list-subtype-bag, 
subtype_rel_self, 
list_wf, 
empty-bag_wf, 
bag_union_cons_lemma, 
assert_functionality_wrt_uiff, 
bag-append_wf, 
band_wf, 
bag-null-append, 
cons-bag_wf, 
equal-wf-T-base, 
assert-bag-null, 
bag-member-cons, 
iff_transitivity, 
iff_weakening_uiff, 
assert_of_band, 
equal_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
thin, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
cumulativity, 
hypothesisEquality, 
hypothesis, 
imageElimination, 
sqequalRule, 
lambdaEquality, 
functionEquality, 
dependent_functionElimination, 
independent_functionElimination, 
lambdaFormation, 
because_Cache, 
productElimination, 
promote_hyp, 
rename, 
applyEquality, 
independent_isectElimination, 
natural_numberEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
hyp_replacement, 
equalitySymmetry, 
Error :applyLambdaEquality, 
imageMemberEquality, 
baseClosed, 
equalityTransitivity, 
universeEquality, 
productEquality, 
independent_pairFormation, 
inlFormation, 
comment, 
inrFormation
Latex:
\mforall{}[T:Type].  \mforall{}[bbs:bag(bag(T))].
    \muparrow{}bag-null(bag-union(bbs))  supposing  \mforall{}bs:bag(T).  (bs  \mdownarrow{}\mmember{}  bbs  {}\mRightarrow{}  (\muparrow{}bag-null(bs)))
Date html generated:
2016_10_25-AM-10_28_50
Last ObjectModification:
2016_07_12-AM-06_45_09
Theory : bags
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