Nuprl Lemma : l_sum_as_accum

[T:Type]. ∀[f:T ⟶ ℤ]. ∀[L:T List]. ∀[x:ℤ].
  (accumulate (with value and list item a):
   over list:
   with starting value:
    x) l_sum(map(λx.f[x];L)) x)


Definitions occuring in Statement :  l_sum: l_sum(L) map: map(f;as) list_accum: list_accum list: List uall: [x:A]. B[x] so_apply: x[s] lambda: λx.A[x] function: x:A ⟶ B[x] add: m int: 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] 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)
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 equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases list_accum_nil_lemma map_nil_lemma l_sum_nil_lemma zero-add 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 list_accum_cons_lemma map_cons_lemma l_sum_cons_lemma l_sum_wf map_wf 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 functionEquality universeEquality

\mforall{}[T:Type].  \mforall{}[f:T  {}\mrightarrow{}  \mBbbZ{}].  \mforall{}[L:T  List].  \mforall{}[x:\mBbbZ{}].
    (accumulate  (with  value  s  and  list  item  a):
        s  +  f[a]
      over  list:
      with  starting  value:
        x)  \msim{}  l\_sum(map(\mlambda{}x.f[x];L))  +  x)

Date html generated: 2017_04_17-AM-08_40_05
Last ObjectModification: 2017_02_27-PM-04_59_48

Theory : list_1

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