Nuprl Lemma : empty-bag-union
∀T:Type. ∀bs:bag(bag(T)). ((bag-union(bs) = {} ∈ bag(T)) ⇒ (bs ~ bag-rep(#(bs);{})))
Proof
Definitions occuring in Statement :
bag-rep: bag-rep(n;x),
bag-union: bag-union(bbs),
bag-size: #(bs),
empty-bag: {},
bag: bag(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
bag-rep: bag-rep(n;x),
concat: concat(ll),
cons-bag: x.b,
empty-bag: {},
nat: ℕ,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
le: A ≤ B,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
bag-size: #(bs),
bag: bag(T),
quotient: x,y:A//B[x; y],
true: True,
bag-union: bag-union(bbs),
bag-append: as + bs,
iff: P ⇐⇒ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
rev_implies: P ⇐ Q
Lemmas referenced :
equal-wf-T-base,
bag_wf,
bag-union_wf,
nil_wf,
top_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
list_wf,
reduce_wf,
append_wf,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
reduce_nil_lemma,
length_of_nil_lemma,
primrec0_lemma,
equal-wf-base,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
reduce_cons_lemma,
length_of_cons_lemma,
primrec-unroll,
length_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
non_neg_length,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
append_is_nil,
null_nil_lemma,
btrue_wf,
and_wf,
null_wf,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse,
add-subtract-cancel,
permutation_wf,
permutation_weakening,
subtype_rel_list,
bag-subtype-list,
member_wf,
squash_wf,
true_wf,
permutation-length,
list_extensionality,
l_member_wf,
bag-append_wf,
empty-bag_wf,
member-implies-null-eq-bfalse,
cons_wf,
bag-append-is-empty,
cons_member,
equal-empty-bag,
equal-nil-sq-nil,
select_wf,
select_member,
lelt_wf,
decidable__lt,
member-permutation,
l_member_subtype
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
baseClosed,
universeEquality,
sqequalRule,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
applyEquality,
because_Cache,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
instantiate,
imageElimination,
equalityElimination,
pointwiseFunctionality,
pertypeElimination,
productEquality,
imageMemberEquality,
comment,
hyp_replacement
Latex:
\mforall{}T:Type. \mforall{}bs:bag(bag(T)). ((bag-union(bs) = \{\}) {}\mRightarrow{} (bs \msim{} bag-rep(\#(bs);\{\})))
Date html generated:
2017_10_01-AM-08_52_10
Last ObjectModification:
2017_07_26-PM-04_33_50
Theory : bags
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