Nuprl Lemma : filter-as-accum-aux

[A:Type]. ∀[p:A ⟶ 𝔹]. ∀[L:A List]. ∀[X:Top List].
  (filter(p;rev(L)) accumulate (with value and list item x):
                           if p[x] then [x a] else fi 
                          over list:
                          with starting value:


Definitions occuring in Statement :  filter: filter(P;l) reverse: rev(as) append: as bs list_accum: list_accum cons: [a b] list: List ifthenelse: if then else fi  bool: 𝔹 uall: [x:A]. B[x] top: Top so_apply: x[s] function: x:A ⟶ B[x] universe: Type sqequal: t
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T all: x:A. B[x] nat: implies:  Q false: False ge: i ≥  uimplies: supposing a satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] not: ¬A top: Top and: P ∧ Q prop: subtype_rel: A ⊆B guard: {T} or: P ∨ Q so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] append: as bs so_lambda: so_lambda(x,y,z.t[x; y; z]) so_apply: x[s1;s2;s3] cons: [a b] colength: colength(L) 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) bool: 𝔹 unit: Unit btrue: tt uiff: uiff(P;Q) ifthenelse: if then else fi  bfalse: ff bnot: ¬bb assert: b
Lemmas referenced :  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 list_wf top_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases reverse_nil_lemma list_accum_nil_lemma filter_nil_lemma list_ind_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 reverse-cons list_accum_cons_lemma filter_append reverse_wf cons_wf nil_wf filter_cons_lemma bool_wf eqtt_to_assert append_assoc_sq list_ind_cons_lemma eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot append-nil filter_wf5 subtype_rel_dep_function l_member_wf subtype_rel_self set_wf subtype_rel_list
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination sqequalAxiom cumulativity applyEquality because_Cache unionElimination promote_hyp hypothesis_subsumption productElimination equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination functionExtensionality equalityElimination setEquality functionEquality universeEquality

\mforall{}[A:Type].  \mforall{}[p:A  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[L:A  List].  \mforall{}[X:Top  List].
    (filter(p;rev(L))  @  X  \msim{}  accumulate  (with  value  a  and  list  item  x):
                                                      if  p[x]  then  [x  /  a]  else  a  fi 
                                                    over  list:
                                                    with  starting  value:

Date html generated: 2018_05_21-PM-06_52_18
Last ObjectModification: 2017_07_26-PM-04_58_22

Theory : general

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