Nuprl Lemma : non-empty-bag-union-of-list
∀[T:Type]. ∀L:bag(T) List. (0 < #(bag-union(L)) ⇐⇒ (∃b∈L. 0 < #(b)))
Proof
Definitions occuring in Statement :
bag-union: bag-union(bbs),
bag-size: #(bs),
bag: bag(T),
l_exists: (∃x∈L. P[x]),
list: T List,
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nat: ℕ,
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
bag-size: #(bs),
bag-union: bag-union(bbs),
concat: concat(ll),
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
false: False,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
rev_implies: P ⇐ Q,
bag-append: as + bs,
or: P ∨ Q,
decidable: Dec(P),
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
nat_plus: ℕ+,
uiff: uiff(P;Q)
Lemmas referenced :
list_induction,
bag_wf,
iff_wf,
less_than_wf,
bag-size_wf,
bag-union_wf,
list-subtype-bag,
subtype_rel_self,
nat_wf,
l_exists_wf,
l_member_wf,
list_wf,
reduce_nil_lemma,
length_of_nil_lemma,
false_wf,
l_exists_nil,
l_exists_wf_nil,
reduce_cons_lemma,
bag-size-append,
l_exists_cons,
cons_wf,
or_wf,
decidable__lt,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
intformless_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
add_nat_plus,
nat_plus_wf,
nat_plus_properties,
add-is-int-iff,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
natural_numberEquality,
applyEquality,
because_Cache,
independent_isectElimination,
setElimination,
rename,
setEquality,
independent_functionElimination,
dependent_functionElimination,
universeEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
imageElimination,
productElimination,
addLevel,
impliesFunctionality,
addEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
applyLambdaEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
dependent_set_memberEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
inlFormation,
inrFormation
Latex:
\mforall{}[T:Type]. \mforall{}L:bag(T) List. (0 < \#(bag-union(L)) \mLeftarrow{}{}\mRightarrow{} (\mexists{}b\mmember{}L. 0 < \#(b)))
Date html generated:
2017_10_01-AM-08_47_04
Last ObjectModification:
2017_07_26-PM-04_31_44
Theory : bags
Home
Index