Nuprl Lemma : assert-bag-null
∀[T:Type]. ∀[bs:bag(T)]. uiff(↑bag-null(bs);bs = {} ∈ bag(T))
Proof
Definitions occuring in Statement :
bag-null: bag-null(bs),
empty-bag: {},
bag: bag(T),
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bag: bag(T),
quotient: x,y:A//B[x; y],
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
true: True,
subtype_rel: A ⊆r B,
bag-null: bag-null(bs),
empty-bag: {},
so_lambda: λ2x.t[x],
so_apply: x[s],
assert: ↑b,
ifthenelse: if b then t else f fi ,
null: null(as),
nil: [],
it: ⋅,
btrue: tt
Lemmas referenced :
assert_wf,
bag-null_wf,
assert_witness,
equal-wf-T-base,
bag_wf,
list_wf,
permutation_wf,
permutation_weakening,
equal-wf-base,
equal_wf,
squash_wf,
true_wf,
quotient-member-eq,
permutation-equiv,
empty-bag_wf,
assert_of_null,
length_wf_nat,
nat_wf,
subtype_rel_set,
list-subtype-bag
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
cumulativity,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
baseClosed,
sqequalRule,
productElimination,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
because_Cache,
universeEquality,
promote_hyp,
lambdaFormation,
independent_isectElimination,
pointwiseFunctionality,
pertypeElimination,
productEquality,
applyEquality,
lambdaEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
dependent_set_memberEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[bs:bag(T)]. uiff(\muparrow{}bag-null(bs);bs = \{\})
Date html generated:
2017_10_01-AM-08_45_38
Last ObjectModification:
2017_07_26-PM-04_30_50
Theory : bags
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