### 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: `s = 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: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` subtype_rel: `A ⊆r 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: `λ2x y.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: `P `⇐⇒` Q` rev_implies: `P `` 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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