### 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)`
`  ∧ (Σ(f[x] | x < m) ∈ ℤ)`
`  ∧ (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]` and: `P ∧ Q` member: `t ∈ T` 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` and: `P ∧ Q` cand: `A c∧ B` subtype_rel: `A ⊆r B` nat: `ℕ` uimplies: `b supposing a` so_lambda: `λ2x.t[x]` so_apply: `x[s]` 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]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` le: `A ≤ B` less_than: `a < b` uiff: `uiff(P;Q)`
Lemmas referenced :  nat_wf add-member-int_seg1 le_wf int_term_value_subtract_lemma int_formula_prop_le_lemma itermSubtract_wf intformle_wf decidable__le subtract_wf int_seg_wf lelt_wf int_formula_prop_wf int_term_value_constant_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermConstant_wf itermAdd_wf itermVar_wf intformless_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__lt nat_properties int_seg_properties int_seg_subtype_nat sum_wf sum_split
Rules used in proof :  cut lemma_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality independent_pairFormation applyEquality natural_numberEquality addEquality setElimination rename independent_isectElimination introduction because_Cache sqequalRule lambdaEquality dependent_set_memberEquality productElimination dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll functionEquality

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)))
\mwedge{}  (\mSigma{}(f[x]  |  x  <  m)  \mmember{}  \mBbbZ{})
\mwedge{}  (\mSigma{}(f[x  +  m]  |  x  <  n  -  m)  \mmember{}  \mBbbZ{}))

Date html generated: 2016_05_14-AM-07_33_27
Last ObjectModification: 2016_01_14-PM-09_54_37

Theory : int_2

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