Nuprl Lemma : bag-combine-null
∀[A,B:Type]. ∀[f:A ⟶ bag(B)]. ∀[b:bag(A)]. uiff(↑bag-null(⋃x∈b.f[x]);∀x:A. (x ↓∈ b ⇒ (↑bag-null(f[x]))))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-combine: ⋃x∈bs.f[x],
bag-null: bag-null(bs),
bag: bag(T),
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
prop: ℙ,
so_lambda: λ2x.t[x],
not: ¬A,
false: False,
exists: ∃x:A. B[x],
cand: A c∧ B,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q
Lemmas referenced :
bag-member-empty-iff,
iff_weakening_equal,
true_wf,
squash_wf,
and_wf,
bag-member-combine,
empty-bag-iff-no-member,
bag_wf,
all_wf,
bag-combine_wf,
bag-null_wf,
assert_wf,
assert_witness,
bag-member_wf,
assert-bag-null
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
productElimination,
independent_isectElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
because_Cache,
independent_functionElimination,
functionEquality,
universeEquality,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
imageMemberEquality,
baseClosed,
voidElimination,
imageElimination,
natural_numberEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} bag(B)]. \mforall{}[b:bag(A)].
uiff(\muparrow{}bag-null(\mcup{}x\mmember{}b.f[x]);\mforall{}x:A. (x \mdownarrow{}\mmember{} b {}\mRightarrow{} (\muparrow{}bag-null(f[x]))))
Date html generated:
2016_05_15-PM-02_40_16
Last ObjectModification:
2016_01_16-AM-08_47_48
Theory : bags
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