Nuprl Lemma : bag-size-bag-member
∀[T:Type]. ∀[bs:bag(T)]. (0 < #(bs) ⇐⇒ ↓∃b:T. b ↓∈ bs)
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-size: #(bs),
bag: bag(T),
less_than: a < b,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
squash: ↓T,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
bag-size: #(bs),
bag-member: x ↓∈ bs,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
all: ∀x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
rev_implies: P ⇐ Q,
bag: bag(T),
quotient: x,y:A//B[x; y],
guard: {T},
or: P ∨ Q,
cons: [a / b],
top: Top,
decidable: Dec(P),
uiff: uiff(P;Q),
subtract: n - m,
true: True
Lemmas referenced :
bag_to_squash_list,
less_than_wf,
bag-size_wf,
select_wf,
false_wf,
select_member,
lelt_wf,
length_wf,
l_member_wf,
list-subtype-bag,
equal_wf,
bag_wf,
squash_wf,
exists_wf,
list_wf,
bag-member_wf,
nat_wf,
member-less_than,
member-permutation,
list-cases,
length_of_nil_lemma,
nil_member,
product_subtype_list,
length_of_cons_lemma,
length_wf_nat,
decidable__lt,
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,
member_wf,
permutation_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
imageElimination,
productElimination,
promote_hyp,
hypothesis,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
natural_numberEquality,
cumulativity,
applyEquality,
because_Cache,
sqequalRule,
rename,
dependent_pairFormation,
independent_isectElimination,
dependent_functionElimination,
dependent_set_memberEquality,
lambdaEquality,
productEquality,
imageMemberEquality,
baseClosed,
setElimination,
independent_pairEquality,
isect_memberEquality,
universeEquality,
pertypeElimination,
independent_functionElimination,
unionElimination,
voidElimination,
hypothesis_subsumption,
voidEquality,
addEquality,
intEquality,
minusEquality,
equalityTransitivity
Latex:
\mforall{}[T:Type]. \mforall{}[bs:bag(T)]. (0 < \#(bs) \mLeftarrow{}{}\mRightarrow{} \mdownarrow{}\mexists{}b:T. b \mdownarrow{}\mmember{} bs)
Date html generated:
2017_10_01-AM-08_54_49
Last ObjectModification:
2017_07_26-PM-04_36_40
Theory : bags
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