Nuprl Lemma : list_accum_filter

∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[l:T List]. ∀[f,y:Top].
  (accumulate (with value a and list item b):
    f[a;b]
   over list:
     filter(λb.P[b];l)
   with starting value:
    y) ~ accumulate (with value a and list item b):
          if P[b] then f[a;b] else a fi 
         over list:
           l
         with starting value:
          y))


Proof



Definitions occuring in Statement :  filter: filter(P;l),  list_accum: list_accum,  list: T List,  ifthenelse: if b then t else f fi ,  bool: 𝔹,  uall: ∀[x:A]. B[x],  top: Top,  so_apply: x[s1;s2],  so_apply: x[s],  lambda: λx.A[x],  function: x:A ⟶ B[x],  universe: Type,  sqequal: s ~ t
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  member: t ∈ T,  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,  top: Top,  and: P ∧ Q,  prop: ℙ,  subtype_rel: A ⊆r B,  guard: {T},  or: P ∨ Q,  so_lambda: λ2x y.t[x; y],  so_apply: x[s1;s2],  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 b then t else f 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,  top_wf,  equal-wf-T-base,  nat_wf,  colength_wf_list,  less_than_transitivity1,  less_than_irreflexivity,  list-cases,  filter_nil_lemma,  list_accum_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,  filter_cons_lemma,  list_accum_cons_lemma,  bool_wf,  eqtt_to_assert,  eqff_to_assert,  bool_cases_sqequal,  bool_subtype_base,  assert-bnot,  list_wf
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,  functionEquality,  universeEquality
Latex:
\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[l:T  List].  \mforall{}[f,y:Top].
    (accumulate  (with  value  a  and  list  item  b):
        f[a;b]
      over  list:
          filter(\mlambda{}b.P[b];l)
      with  starting  value:
        y)  \msim{}  accumulate  (with  value  a  and  list  item  b):
                    if  P[b]  then  f[a;b]  else  a  fi 
                  over  list:
                      l
                  with  starting  value:
                    y))


Date html generated: 2017_04_14-AM-09_24_31
Last ObjectModification: 2017_02_27-PM-03_59_09

Theory : list_1


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