Nuprl Lemma : rsum-split

[n,m:ℤ]. ∀[x:{n..m 1-} ⟶ ℝ]. ∀[k:ℤ].
  {x[i] n≤i≤m} {x[i] n≤i≤k} + Σ{x[i] 1≤i≤m})) 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] add: m natural_number: $n int:
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a rsum: Σ{x[k] n≤k≤m} so_lambda: λ2x.t[x] so_apply: x[s] int_seg: {i..j-} lelt: i ≤ j < k and: P ∧ Q 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 top: Top prop: callbyvalueall: callbyvalueall has-value: (a)↓ has-valueall: has-valueall(a) subtype_rel: A ⊆B uiff: uiff(P;Q) rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :  radd-list-append req_inversion req_functionality req_weakening from-upto-split map_append_sq append_wf subtype_rel_self list-subtype-bag radd-list_wf-bag valueall-type-real-list evalall-reduce from-upto_wf less_than_wf and_wf map_wf real-valueall-type list-valueall-type list_wf valueall-type-has-valueall int-value-type value-type-has-value real_wf le_wf decidable__le lelt_wf int_formula_prop_wf int_formula_prop_le_lemma 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 intformle_wf itermConstant_wf itermAdd_wf itermVar_wf intformless_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__lt radd_wf int_seg_wf rsum_wf req_witness
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality sqequalRule lambdaEquality applyEquality addEquality natural_numberEquality hypothesis setElimination rename dependent_set_memberEquality productElimination independent_pairFormation dependent_functionElimination unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll because_Cache independent_functionElimination equalityTransitivity equalitySymmetry functionEquality setEquality callbyvalueReduce productEquality lambdaFormation

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

Date html generated: 2016_05_18-AM-07_45_36
Last ObjectModification: 2016_01_17-AM-02_08_00

Theory : reals

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