Nuprl Lemma : sum-ite

[k:ℕ]. ∀[f,g:ℕk ⟶ ℤ]. ∀[p:ℕk ⟶ 𝔹].
  (if p[i] then f[i] g[i] else f[i] fi  i < k) (f[i] i < k) + Σ(if p[i] then g[i] else fi  i < k)) ∈ ℤ)


Proof




Definitions occuring in Statement :  sum: Σ(f[x] x < k) int_seg: {i..j-} nat: ifthenelse: if then else fi  bool: 𝔹 uall: [x:A]. B[x] so_apply: x[s] function: x:A ⟶ B[x] add: m natural_number: $n int: equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T so_lambda: λ2x.t[x] so_apply: x[s] nat: all: x:A. B[x] 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: decidable: Dec(P) or: P ∨ Q subtype_rel: A ⊆B le: A ≤ B less_than': less_than'(a;b) bool: 𝔹 unit: Unit it: btrue: tt uiff: uiff(P;Q) ifthenelse: if then else fi  bfalse: ff iff: ⇐⇒ Q rev_implies:  Q int_seg: {i..j-} lelt: i ≤ j < k sq_type: SQType(T) guard: {T} bnot: ¬bb assert: b
Lemmas referenced :  sum-as-primrec ifthenelse_wf int_seg_wf 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 bool_wf primrec0_lemma decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma subtype_rel_dep_function int_seg_subtype false_wf subtype_rel_self primrec-unroll eq_int_wf uiff_transitivity equal-wf-base int_subtype_base assert_wf eqtt_to_assert assert_of_eq_int intformeq_wf int_formula_prop_eq_lemma iff_transitivity bnot_wf not_wf iff_weakening_uiff eqff_to_assert assert_of_bnot decidable__lt lelt_wf decidable__equal_int add-is-int-iff itermAdd_wf int_term_value_add_lemma equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot nat_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality lambdaEquality applyEquality functionExtensionality natural_numberEquality setElimination rename because_Cache hypothesis intEquality addEquality lambdaFormation intWeakElimination independent_isectElimination dependent_pairFormation int_eqEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination axiomEquality functionEquality unionElimination equalityElimination baseApply closedConclusion baseClosed productElimination impliesFunctionality dependent_set_memberEquality equalityTransitivity equalitySymmetry pointwiseFunctionality promote_hyp instantiate cumulativity

Latex:
\mforall{}[k:\mBbbN{}].  \mforall{}[f,g:\mBbbN{}k  {}\mrightarrow{}  \mBbbZ{}].  \mforall{}[p:\mBbbN{}k  {}\mrightarrow{}  \mBbbB{}].
    (\mSigma{}(if  p[i]  then  f[i]  +  g[i]  else  f[i]  fi    |  i  <  k)
    =  (\mSigma{}(f[i]  |  i  <  k)  +  \mSigma{}(if  p[i]  then  g[i]  else  0  fi    |  i  <  k)))



Date html generated: 2017_04_14-AM-09_20_36
Last ObjectModification: 2017_02_27-PM-03_57_06

Theory : int_2


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