Nuprl Lemma : bag-member-iff-size
∀[T:Type]. ∀[bs:bag(T)]. uiff(↓∃x:T. x ↓∈ bs;0 < #(bs))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-size: #(bs),
bag: bag(T),
less_than: a < b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
exists: ∃x:A. B[x],
bag-member: x ↓∈ bs,
bag-size: #(bs),
all: ∀x:A. B[x],
or: P ∨ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
false: False,
cons: [a / b],
top: Top,
guard: {T},
le: A ≤ B,
decidable: Dec(P),
not: ¬A,
rev_implies: P ⇐ Q,
subtract: n - m,
less_than': less_than'(a;b),
true: True,
bag: bag(T),
quotient: x,y:A//B[x; y],
less_than: a < b,
length: ||as||,
list_ind: list_ind,
nil: [],
it: ⋅,
cand: A c∧ B
Lemmas referenced :
squash_wf,
exists_wf,
bag-member_wf,
less_than_wf,
bag-size_wf,
nat_wf,
member-less_than,
bag_wf,
list-cases,
length_of_nil_lemma,
nil_member,
product_subtype_list,
length_of_cons_lemma,
length_wf_nat,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
equal_wf,
equal-wf-base,
list_wf,
permutation_wf,
length_wf,
list-subtype-bag,
cons_wf,
l_member_wf,
cons_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalHypSubstitution,
imageElimination,
hypothesis,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
imageMemberEquality,
baseClosed,
natural_numberEquality,
applyEquality,
setElimination,
rename,
productElimination,
independent_pairEquality,
isect_memberEquality,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
unionElimination,
independent_functionElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
voidEquality,
lambdaFormation,
addEquality,
intEquality,
minusEquality,
hyp_replacement,
applyLambdaEquality,
pointwiseFunctionalityForEquality,
functionEquality,
pertypeElimination,
productEquality,
dependent_pairFormation,
inlFormation
Latex:
\mforall{}[T:Type]. \mforall{}[bs:bag(T)]. uiff(\mdownarrow{}\mexists{}x:T. x \mdownarrow{}\mmember{} bs;0 < \#(bs))
Date html generated:
2017_10_01-AM-08_53_21
Last ObjectModification:
2017_07_26-PM-04_35_04
Theory : bags
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