### Nuprl Lemma : sum_split

`∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ]. ∀[m:ℕn + 1].  (Σ(f[x] | x < n) = (Σ(f[x] | x < m) + Σ(f[x + m] | x < n - m)) ∈ ℤ)`

Proof

Definitions occuring in Statement :  sum: `Σ(f[x] | x < k)` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` so_apply: `x[s]` function: `x:A ⟶ B[x]` subtract: `n - m` add: `n + m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nat: `ℕ` subtype_rel: `A ⊆r B` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` int_seg: `{i..j-}` lelt: `i ≤ j < k` guard: `{T}` ge: `i ≥ j ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` less_than: `a < b` uiff: `uiff(P;Q)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T `
Lemmas referenced :  sum-as-primrec int_seg_wf int_seg_subtype_nat false_wf int_seg_properties nat_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermVar_wf itermAdd_wf itermConstant_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_wf lelt_wf subtract_wf decidable__le intformle_wf itermSubtract_wf int_formula_prop_le_lemma int_term_value_subtract_lemma le_wf add-member-int_seg1 primrec_add subtract-add-cancel primrec_wf nat_wf equal_wf intformeq_wf int_formula_prop_eq_lemma ge_wf less_than_wf primrec0_lemma decidable__equal_int eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int add-member-int_seg2 intformimplies_wf int_formual_prop_imp_lemma primrec-unroll
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 addEquality independent_isectElimination independent_pairFormation lambdaFormation dependent_set_memberEquality productElimination dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll equalityTransitivity equalitySymmetry independent_functionElimination axiomEquality functionEquality applyLambdaEquality intWeakElimination equalityElimination promote_hyp instantiate cumulativity

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[f:\mBbbN{}n  {}\mrightarrow{}  \mBbbZ{}].  \mforall{}[m:\mBbbN{}n  +  1].    (\mSigma{}(f[x]  |  x  <  n)  =  (\mSigma{}(f[x]  |  x  <  m)  +  \mSigma{}(f[x  +  m]  |  x  <  n  -  m)))

Date html generated: 2017_04_14-AM-09_21_15
Last ObjectModification: 2017_02_27-PM-03_57_55

Theory : int_2

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