### Nuprl Lemma : bag-mapfilter-fast-eq

`∀[A,B:Type]. ∀[bs:bag(A)]. ∀[P:A ⟶ 𝔹]. ∀[f:{x:A| ↑P[x]}  ⟶ B].`
`  (bag-mapfilter-fast(f;P;bs) = bag-mapfilter(f;P;bs) ∈ bag(B))`

Proof

Definitions occuring in Statement :  bag-mapfilter-fast: `bag-mapfilter-fast(f;P;bs)` bag-mapfilter: `bag-mapfilter(f;P;bs)` bag: `bag(T)` assert: `↑b` 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-mapfilter: `bag-mapfilter(f;P;bs)` so_apply: `x[s]` prop: `ℙ` so_lambda: `λ2x.t[x]` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` uimplies: `b supposing a` guard: `{T}` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` implies: `P `` Q` all: `∀x:A. B[x]` top: `Top` so_apply: `x[s1;s2]` so_lambda: `λ2x y.t[x; y]` bag-mapfilter-fast: `bag-mapfilter-fast(f;P;bs)` exists: `∃x:A. B[x]` nat: `ℕ` false: `False` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` not: `¬A` or: `P ∨ Q` cons: `[a / b]` colength: `colength(L)` decidable: `Dec(P)` nil: `[]` it: `⋅` sq_type: `SQType(T)` less_than: `a < b` less_than': `less_than'(a;b)` bool: `𝔹` unit: `Unit` btrue: `tt` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` bfalse: `ff` empty-bag: `{}` bag-accum: `bag-accum(v,x.f[v; x];init;bs)` cons-bag: `x.b` bnot: `¬bb` assert: `↑b` label: `...\$L... t` bag-append: `as + bs` single-bag: `{x}` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]`
Lemmas referenced :  bag-map-as-accum assert_wf bag-filter_wf equal_wf squash_wf true_wf bag_wf bag-mapfilter-fast_wf iff_weakening_equal bag-filter-as-accum bool_wf set_wf empty-bag_wf cons-bag_wf cons-bag-as-append bag-append_wf bag-append-comm single-bag_wf bag-accum_wf all_wf bag-append-assoc 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 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 eqtt_to_assert list_accum_nil_lemma nil_wf list-subtype-bag bag-append-assoc-comm eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot subtype_rel_list top_wf list_accum_append bag-accum-single list_accum_wf ifthenelse_wf append_wf bag-subtype-list list_ind_cons_lemma list_ind_nil_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin setEquality cumulativity hypothesisEquality applyEquality functionExtensionality hypothesis sqequalRule lambdaEquality imageElimination equalityTransitivity equalitySymmetry because_Cache natural_numberEquality imageMemberEquality baseClosed universeEquality independent_isectElimination productElimination independent_functionElimination functionEquality isect_memberEquality axiomEquality setElimination rename dependent_set_memberEquality lambdaFormation voidElimination voidEquality dependent_functionElimination promote_hyp intWeakElimination dependent_pairFormation int_eqEquality intEquality independent_pairFormation computeAll unionElimination hypothesis_subsumption applyLambdaEquality addEquality instantiate hyp_replacement equalityElimination

Latex:
\mforall{}[A,B:Type].  \mforall{}[bs:bag(A)].  \mforall{}[P:A  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[f:\{x:A|  \muparrow{}P[x]\}    {}\mrightarrow{}  B].
(bag-mapfilter-fast(f;P;bs)  =  bag-mapfilter(f;P;bs))

Date html generated: 2017_10_01-AM-08_58_35
Last ObjectModification: 2017_07_26-PM-04_40_28

Theory : bags

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