Nuprl Lemma : bag-union_wf
∀[T:Type]. ∀[bbs:bag(bag(T))]. (bag-union(bbs) ∈ bag(T))
Proof
Definitions occuring in Statement :
bag-union: bag-union(bbs),
bag: bag(T),
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bag-union: bag-union(bbs),
prop: ℙ,
nat: ℕ,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
or: P ∨ Q,
concat: concat(ll),
empty-bag: {},
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
subtype_rel: A ⊆r B,
bag-append: as + bs,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
bag_wf,
permutation_wf,
list_wf,
istype-universe,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
list-cases,
reduce_nil_lemma,
empty-bag_wf,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
le_wf,
reduce_cons_lemma,
bag-append_wf,
istype-nat,
permutation-invariant,
equal_wf,
cons_wf,
concat_append,
append-nil,
bag-subtype-list,
bag-append-comm,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
bag-append-assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
pertypeElimination,
promote_hyp,
productElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
lambdaFormation_alt,
rename,
universeIsType,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
productIsType,
sqequalBase,
because_Cache,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
instantiate,
universeEquality,
setElimination,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
voidElimination,
independent_pairFormation,
functionIsTypeImplies,
unionElimination,
hypothesis_subsumption,
dependent_set_memberEquality_alt,
applyLambdaEquality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[bbs:bag(bag(T))]. (bag-union(bbs) \mmember{} bag(T))
Date html generated:
2019_10_15-AM-11_00_11
Last ObjectModification:
2018_11_30-AM-09_53_02
Theory : bags
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