Nuprl Lemma : bag-map-as-accum

[A,B:Type]. ∀[f:A ⟶ B]. ∀[bs:bag(A)].  (bag-map(f;bs) bag-accum(b,x.f[x].b;{};bs) ∈ bag(B))


Proof




Definitions occuring in Statement :  bag-accum: bag-accum(v,x.f[v; x];init;bs) bag-map: bag-map(f;bs) cons-bag: x.b empty-bag: {} bag: bag(T) uall: [x:A]. B[x] so_apply: x[s] function: x:A ⟶ B[x] universe: Type equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T squash: T exists: x:A. B[x] all: x:A. B[x] nat: implies:  Q false: False ge: i ≥  uimplies: supposing a satisfiable_int_formula: satisfiable_int_formula(fmla) not: ¬A top: Top and: P ∧ Q prop: subtype_rel: A ⊆B guard: {T} or: P ∨ Q bag-accum: bag-accum(v,x.f[v; x];init;bs) list_accum: list_accum nil: [] it: empty-bag: {} bag-map: bag-map(f;bs) map: map(f;as) list_ind: list_ind cons: [a b] colength: colength(L) so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] decidable: Dec(P) so_lambda: λ2x.t[x] so_apply: x[s] sq_type: SQType(T) less_than: a < b less_than': less_than'(a;b) cons-bag: x.b true: True iff: ⇐⇒ Q rev_implies:  Q bag-append: as bs
Lemmas referenced :  bag_to_squash_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_wf list-cases bag-map_wf nil_wf list-subtype-bag 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 bag-map-cons cons-bag-as-append bag-append-comm single-bag_wf bag_wf cons-bag_wf bag-accum_wf empty-bag_wf bag-append_wf iff_weakening_equal bag-append-assoc-comm list_accum_append subtype_rel_list top_wf squash_wf true_wf all_wf bag-append-assoc bag-accum-single
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache hypothesisEquality imageElimination productElimination promote_hyp hypothesis rename lambdaFormation setElimination intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination axiomEquality cumulativity applyEquality unionElimination hypothesis_subsumption equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate hyp_replacement functionExtensionality functionEquality universeEquality imageMemberEquality

Latex:
\mforall{}[A,B:Type].  \mforall{}[f:A  {}\mrightarrow{}  B].  \mforall{}[bs:bag(A)].    (bag-map(f;bs)  =  bag-accum(b,x.f[x].b;\{\};bs))



Date html generated: 2017_10_01-AM-08_48_16
Last ObjectModification: 2017_07_26-PM-04_32_28

Theory : bags


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