### Nuprl Lemma : bag-filter-union

`∀[T:Type]. ∀[p:T ⟶ 𝔹]. ∀[bbs:bag(bag(T))].`
`  ([x∈bag-union(bbs)|p[x]] = bag-union(bag-map(λb.[x∈b|p[x]];bbs)) ∈ bag({x:T| ↑p[x]} ))`

Proof

Definitions occuring in Statement :  bag-union: `bag-union(bbs)` bag-filter: `[x∈b|p[x]]` bag-map: `bag-map(f;bs)` bag: `bag(T)` assert: `↑b` bool: `𝔹` uall: `∀[x:A]. B[x]` so_apply: `x[s]` set: `{x:A| B[x]} ` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` bag: `bag(T)` so_apply: `x[s]` prop: `ℙ` quotient: `x,y:A//B[x; y]` and: `P ∧ Q` all: `∀x:A. B[x]` implies: `P `` Q` subtype_rel: `A ⊆r B` uimplies: `b supposing a` bag-union: `bag-union(bbs)` bag-filter: `[x∈b|p[x]]` bag-map: `bag-map(f;bs)` concat: `concat(ll)` nat: `ℕ` false: `False` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` 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]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True`
Lemmas referenced :  bag_wf assert_wf list_wf permutation_wf equal_wf equal-wf-base bool_wf subtype_rel_list top_wf bag-subtype-list 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 filter_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 filter_append_sq bag-filter_wf squash_wf true_wf bag-union_wf quotient-member-eq permutation-equiv
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution pointwiseFunctionalityForEquality extract_by_obid isectElimination thin setEquality cumulativity hypothesisEquality applyEquality functionExtensionality hypothesis sqequalRule pertypeElimination productElimination equalityTransitivity equalitySymmetry lambdaFormation because_Cache rename dependent_functionElimination independent_functionElimination productEquality isect_memberEquality axiomEquality functionEquality independent_isectElimination setElimination intWeakElimination natural_numberEquality dependent_pairFormation lambdaEquality int_eqEquality intEquality voidElimination voidEquality independent_pairFormation computeAll sqequalAxiom unionElimination promote_hyp hypothesis_subsumption applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination universeEquality imageMemberEquality

Latex:
\mforall{}[T:Type].  \mforall{}[p:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[bbs:bag(bag(T))].
([x\mmember{}bag-union(bbs)|p[x]]  =  bag-union(bag-map(\mlambda{}b.[x\mmember{}b|p[x]];bbs)))

Date html generated: 2017_10_01-AM-08_46_38
Last ObjectModification: 2017_07_26-PM-04_31_22

Theory : bags

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