Nuprl Lemma : bag-combine-empty-right
∀[T:Type]. ∀[bs:bag(T)]. (⋃x∈bs.{} ~ {})
Proof
Definitions occuring in Statement :
bag-combine: ⋃x∈bs.f[x],
empty-bag: {},
bag: bag(T),
uall: ∀[x:A]. B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
empty-bag: {},
bag-combine: ⋃x∈bs.f[x],
bag-map: bag-map(f;bs),
bag-union: bag-union(bbs),
concat: concat(ll),
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,
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),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3]
Lemmas referenced :
bag-subtype-list,
list_wf,
top_wf,
equal_wf,
bag_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,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
map_nil_lemma,
reduce_nil_lemma,
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,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
map_cons_lemma,
reduce_cons_lemma,
list_ind_nil_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesisEquality,
applyEquality,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesis,
sqequalRule,
isectElimination,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
sqequalAxiom,
cumulativity,
isect_memberEquality,
because_Cache,
universeEquality,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
imageElimination
Latex:
\mforall{}[T:Type]. \mforall{}[bs:bag(T)]. (\mcup{}x\mmember{}bs.\{\} \msim{} \{\})
Date html generated:
2017_10_01-AM-08_47_12
Last ObjectModification:
2017_07_26-PM-04_31_48
Theory : bags
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