### Nuprl Lemma : rsum-triangle-inequality2

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

Proof

Definitions occuring in Statement :  rsum: `Σ{x[k] | n≤k≤m}` rleq: `x ≤ y` rabs: `|x|` rsub: `x - y` radd: `a + b` real: `ℝ` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` so_apply: `x[s]` 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` rleq: `x ≤ y` rnonneg: `rnonneg(x)` all: `∀x:A. B[x]` le: `A ≤ B` and: `P ∧ Q` not: `¬A` implies: `P `` Q` false: `False` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` prop: `ℙ` int_seg: `{i..j-}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` rev_uimplies: `rev_uimplies(P;Q)` rge: `x ≥ y` guard: `{T}` uiff: `uiff(P;Q)`
Lemmas referenced :  req_weakening radd_comm rabs_functionality rsum_functionality2 rleq_functionality rsum-triangle-inequality1 rleq_functionality_wrt_implies rleq_weakening_equal le_wf 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 real_wf nat_plus_wf int_seg_wf radd_wf rabs_wf rsum_wf rsub_wf less_than'_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule sqequalHypSubstitution lambdaEquality dependent_functionElimination thin hypothesisEquality productElimination independent_pairEquality because_Cache lemma_by_obid isectElimination applyEquality hypothesis addEquality natural_numberEquality minusEquality axiomEquality equalityTransitivity equalitySymmetry functionEquality isect_memberEquality intEquality voidElimination dependent_set_memberEquality independent_pairFormation unionElimination independent_isectElimination dependent_pairFormation int_eqEquality voidEquality computeAll lambdaFormation

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

Date html generated: 2016_05_18-AM-07_48_30
Last ObjectModification: 2016_01_17-AM-02_08_57

Theory : reals

Home Index