Nuprl Lemma : bag-union-bagp
∀[T:Type]. ∀[bbs:bag(bag(T))]. (0 < #([b∈bbs|0 <z #(b)]) ⇒ (bag-union(bbs) ∈ T Bag+))
Proof
Definitions occuring in Statement :
bag-union: bag-union(bbs),
bagp: T Bag+,
bag-size: #(bs),
bag-filter: [x∈b|p[x]],
bag: bag(T),
lt_int: i <z j,
less_than: a < b,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
bagp: T Bag+,
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
empty-bag: {},
all: ∀x:A. B[x],
top: Top,
cons-bag: x.b,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
decidable: Dec(P),
or: P ∨ Q,
nat: ℕ,
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nat_plus: ℕ+,
less_than: a < b
Lemmas referenced :
bag-union_wf,
bag_to_squash_list,
less_than_wf,
bag-size_wf,
bag_wf,
assert_wf,
lt_int_wf,
bag-filter_wf,
list_induction,
list-subtype-bag,
list_wf,
bag_filter_empty_lemma,
bag_union_empty_lemma,
bag_size_empty_lemma,
bag_filter_cons_lemma,
bag_union_cons_lemma,
bag-size-append,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
bag_size_cons_lemma,
decidable__lt,
subtype_rel_self,
nat_wf,
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,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
add_nat_plus,
nat_plus_wf,
nat_plus_properties,
add-is-int-iff,
intformeq_wf,
int_formula_prop_eq_lemma,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
dependent_set_memberEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
because_Cache,
imageElimination,
productElimination,
promote_hyp,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
natural_numberEquality,
setEquality,
applyEquality,
sqequalRule,
lambdaEquality,
rename,
functionEquality,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
addEquality,
setElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
instantiate,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed,
axiomEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[bbs:bag(bag(T))]. (0 < \#([b\mmember{}bbs|0 <z \#(b)]) {}\mRightarrow{} (bag-union(bbs) \mmember{} T Bag\msupplus{}))
Date html generated:
2017_10_01-AM-08_46_29
Last ObjectModification:
2017_07_26-PM-04_31_18
Theory : bags
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