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
Home
Index