Nuprl Lemma : rsum-split-shift

[k,n,m:ℤ]. ∀[x:{n..m 1-} ⟶ ℝ].
  {x[i] n≤i≤m} {x[i] n≤i≤k} + Σ{x[k 1] 0≤i≤1})) supposing ((k ≤ m) and (n ≤ k))


Definitions occuring in Statement :  rsum: Σ{x[k] n≤k≤m} req: y radd: b real: int_seg: {i..j-} uimplies: supposing a uall: [x:A]. B[x] so_apply: x[s] le: A ≤ B function: x:A ⟶ B[x] subtract: m add: m natural_number: $n int:
Definitions unfolded in proof :  uall: [x:A]. B[x] uimplies: supposing a member: t ∈ T prop: top: Top 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:  Q not: ¬A sq_type: SQType(T) guard: {T}
Lemmas referenced :  rsum-shift add-swap int_formula_prop_wf int_term_value_var_lemma int_term_value_add_lemma int_term_value_subtract_lemma int_term_value_constant_lemma int_formula_prop_eq_lemma int_formula_prop_not_lemma itermVar_wf itermAdd_wf itermSubtract_wf itermConstant_wf intformeq_wf intformnot_wf satisfiable-full-omega-tt decidable__equal_int int_subtype_base subtype_base_sq real_wf int_seg_wf le_wf rsum-split
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality independent_isectElimination hypothesis functionEquality addEquality natural_numberEquality because_Cache intEquality isect_memberEquality voidElimination voidEquality sqequalRule instantiate dependent_functionElimination unionElimination dependent_pairFormation lambdaEquality int_eqEquality computeAll equalityTransitivity equalitySymmetry independent_functionElimination

\mforall{}[k,n,m:\mBbbZ{}].  \mforall{}[x:\{n..m  +  1\msupminus{}\}  {}\mrightarrow{}  \mBbbR{}].
    (\mSigma{}\{x[i]  |  n\mleq{}i\mleq{}m\}  =  (\mSigma{}\{x[i]  |  n\mleq{}i\mleq{}k\}  +  \mSigma{}\{x[k  +  i  +  1]  |  0\mleq{}i\mleq{}m  -  k  +  1\}))  supposing 
          ((k  \mleq{}  m)  and 
          (n  \mleq{}  k))

Date html generated: 2016_05_18-AM-07_45_50
Last ObjectModification: 2016_01_17-AM-02_08_25

Theory : reals

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