Nuprl Lemma : rsum-difference2

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


Definitions occuring in Statement :  rsum: Σ{x[k] n≤k≤m} rsub: y req: y real: int_seg: {i..j-} uimplies: supposing a uall: [x:A]. B[x] so_apply: x[s] le: A ≤ B all: x:A. B[x] 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 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: uiff: uiff(P;Q) rev_uimplies: rev_uimplies(P;Q) rsub: y itermConstant: "const" req_int_terms: t1 ≡ t2 real_term_value: real_term_value(f;t) int_term_ind: int_term_ind itermSubtract: left (-) right itermAdd: left (+) right itermVar: vvar itermMinus: "-"num
Lemmas referenced :  rsum-split req_witness rsub_wf rsum_wf int_seg_wf decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermVar_wf itermAdd_wf itermConstant_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_wf lelt_wf decidable__le all_wf req_wf le_wf real_wf radd_wf rminus_wf req_functionality rsub_functionality req_weakening uiff_transitivity req_inversion radd-assoc radd-ac real_term_polynomial itermSubtract_wf itermMinus_wf int-to-real_wf req-iff-rsub-is-0 radd_functionality rminus_functionality rsum_functionality2
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality independent_isectElimination hypothesis sqequalRule lambdaEquality applyEquality functionExtensionality because_Cache addEquality natural_numberEquality setElimination rename dependent_set_memberEquality productElimination independent_pairFormation dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll independent_functionElimination equalityTransitivity equalitySymmetry functionEquality lambdaFormation lemma_by_obid

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

Date html generated: 2017_10_03-AM-08_58_52
Last ObjectModification: 2017_07_28-AM-07_38_30

Theory : reals

Home Index