### 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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