### Nuprl Lemma : rsum-split

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

Proof

Definitions occuring in Statement :  rsum: `Σ{x[k] | n≤k≤m}` req: `x = y` radd: `a + b` real: `ℝ` int_seg: `{i..j-}` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` le: `A ≤ B` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b 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: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` callbyvalueall: callbyvalueall has-value: `(a)↓` has-valueall: `has-valueall(a)` subtype_rel: `A ⊆r 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

Latex:
\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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