### Nuprl Lemma : bag-combine-filter

`∀[A,B:Type]. ∀[p:A ⟶ 𝔹]. ∀[f:{a:A| ↑p[a]}  ⟶ bag(B)]. ∀[ba:bag(A)].`
`  (⋃a∈[a∈ba|p[a]].f[a] = ⋃a∈ba.if p[a] then f[a] else {} fi  ∈ bag(B))`

Proof

Definitions occuring in Statement :  bag-combine: `⋃x∈bs.f[x]` bag-filter: `[x∈b|p[x]]` empty-bag: `{}` bag: `bag(T)` assert: `↑b` ifthenelse: `if b then t else f fi ` bool: `𝔹` uall: `∀[x:A]. B[x]` so_apply: `x[s]` set: `{x:A| B[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)` quotient: `x,y:A//B[x; y]` and: `P ∧ Q` all: `∀x:A. B[x]` implies: `P `` Q` prop: `ℙ` so_apply: `x[s]` cand: `A c∧ B` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` uimplies: `b supposing a` bag-filter: `[x∈b|p[x]]` 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 ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` subtype_rel: `A ⊆r B` guard: `{T}` or: `P ∨ Q` cons: `[a / b]` colength: `colength(L)` 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)` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` bag-append: `as + bs` true: `True`
Lemmas referenced :  bag_wf list_wf permutation_wf equal_wf equal-wf-base assert_wf bool_wf quotient-member-eq permutation-equiv 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 filter_nil_lemma 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 filter_cons_lemma map_cons_lemma reduce_cons_lemma empty-bag_wf eqtt_to_assert eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot list_ind_nil_lemma bag-append_wf squash_wf true_wf bag-combine_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution pointwiseFunctionalityForEquality extract_by_obid isectElimination thin cumulativity hypothesisEquality hypothesis sqequalRule pertypeElimination productElimination equalityTransitivity equalitySymmetry lambdaFormation because_Cache rename dependent_functionElimination independent_functionElimination productEquality isect_memberEquality axiomEquality functionEquality setEquality applyEquality functionExtensionality lambdaEquality independent_isectElimination independent_pairFormation hyp_replacement applyLambdaEquality setElimination intWeakElimination natural_numberEquality dependent_pairFormation int_eqEquality intEquality voidElimination voidEquality computeAll unionElimination promote_hyp hypothesis_subsumption dependent_set_memberEquality addEquality baseClosed instantiate imageElimination equalityElimination universeEquality imageMemberEquality

Latex:
\mforall{}[A,B:Type].  \mforall{}[p:A  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[f:\{a:A|  \muparrow{}p[a]\}    {}\mrightarrow{}  bag(B)].  \mforall{}[ba:bag(A)].
(\mcup{}a\mmember{}[a\mmember{}ba|p[a]].f[a]  =  \mcup{}a\mmember{}ba.if  p[a]  then  f[a]  else  \{\}  fi  )

Date html generated: 2017_10_01-AM-08_47_29
Last ObjectModification: 2017_07_26-PM-04_31_59

Theory : bags

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