### Nuprl Lemma : sum_split_first

`∀[n:ℕ+]. ∀[f:ℕn ⟶ ℤ].  (Σ(f[x] | x < n) = (f[0] + Σ(f[x + 1] | x < n - 1)) ∈ ℤ)`

Proof

Definitions occuring in Statement :  sum: `Σ(f[x] | x < k)` int_seg: `{i..j-}` nat_plus: `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` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` nat_plus: `ℕ+` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` so_lambda: `λ2x.t[x]` so_apply: `x[s]` less_than: `a < b`
Lemmas referenced :  nat_plus_wf int_seg_wf sum1 lelt_wf int_formula_prop_wf int_term_value_var_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermAdd_wf itermConstant_wf intformless_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__lt nat_plus_properties false_wf nat_plus_subtype_nat sum_split
Rules used in proof :  cut lemma_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality applyEquality sqequalRule dependent_set_memberEquality natural_numberEquality independent_pairFormation lambdaFormation setElimination rename dependent_functionElimination addEquality unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll productElimination because_Cache functionEquality

Latex:
\mforall{}[n:\mBbbN{}\msupplus{}].  \mforall{}[f:\mBbbN{}n  {}\mrightarrow{}  \mBbbZ{}].    (\mSigma{}(f[x]  |  x  <  n)  =  (f[0]  +  \mSigma{}(f[x  +  1]  |  x  <  n  -  1)))

Date html generated: 2016_05_14-AM-07_33_30
Last ObjectModification: 2016_01_14-PM-09_54_22

Theory : int_2

Home Index