### Nuprl Lemma : bag-separate-merge

`∀[as,bs:Top List].  (bag-separate(bag-merge(as;bs)) ~ <as, bs>)`

Proof

Definitions occuring in Statement :  bag-separate: `bag-separate(bs)` bag-merge: `bag-merge(as;bs)` list: `T List` uall: `∀[x:A]. B[x]` top: `Top` pair: `<a, b>` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` bag-merge: `bag-merge(as;bs)` bag-separate: `bag-separate(bs)` bag-mapfilter: `bag-mapfilter(f;P;bs)` bag-map: `bag-map(f;bs)` bag-append: `as + bs` bag-filter: `[x∈b|p[x]]` top: `Top` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` and: `P ∧ Q` prop: `ℙ` subtype_rel: `A ⊆r B` 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)` isl: `isl(x)` ifthenelse: `if b then t else f fi ` bfalse: `ff` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` btrue: `tt` outl: `outl(x)` bnot: `¬bb` outr: `outr(x)`
Lemmas referenced :  list_wf top_wf map_append_sq filter_append_sq 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 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 equal_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 filter_cons_lemma list_ind_nil_lemma list_ind_cons_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule hypothesis sqequalAxiom extract_by_obid sqequalHypSubstitution isectElimination thin isect_memberEquality hypothesisEquality because_Cache voidElimination voidEquality lambdaFormation setElimination rename intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination independent_pairFormation computeAll independent_functionElimination applyEquality unionElimination promote_hyp hypothesis_subsumption productElimination equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate cumulativity imageElimination

Latex:
\mforall{}[as,bs:Top  List].    (bag-separate(bag-merge(as;bs))  \msim{}  <as,  bs>)

Date html generated: 2017_10_01-AM-08_52_46
Last ObjectModification: 2017_07_26-PM-04_34_14

Theory : bags

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