Nuprl Lemma : sum_split1

`∀[n:ℕ+]. ∀[f:ℕn ⟶ ℤ].  (Σ(f[x] | x < n) = (Σ(f[x] | x < n - 1) + f[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` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nat_plus: `ℕ+` nat: `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]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` less_than: `a < b` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T `
Lemmas referenced :  sum-as-primrec nat_plus_subtype_nat int_seg_wf subtract_wf nat_plus_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf le_wf decidable__lt lelt_wf primrec-unroll eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int intformeq_wf int_formula_prop_eq_lemma eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int primrec_wf decidable__equal_int nat_plus_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality applyEquality hypothesis lambdaEquality functionExtensionality natural_numberEquality setElimination rename because_Cache dependent_set_memberEquality dependent_functionElimination unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll productElimination lambdaFormation equalityElimination equalityTransitivity equalitySymmetry promote_hyp instantiate cumulativity independent_functionElimination addEquality functionEquality axiomEquality

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

Date html generated: 2017_04_14-AM-09_21_19
Last ObjectModification: 2017_02_27-PM-03_57_23

Theory : int_2

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