### Nuprl Lemma : rsum_functionality_wrt_rleq

`∀[n,m:ℤ]. ∀[x,y:{n..m + 1-} ⟶ ℝ].  Σ{x[k] | n≤k≤m} ≤ Σ{y[k] | n≤k≤m} supposing x[k] ≤ y[k] for k ∈ [n,m]`

Proof

Definitions occuring in Statement :  pointwise-rleq: `x[k] ≤ y[k] for k ∈ [n,m]` rsum: `Σ{x[k] | n≤k≤m}` rleq: `x ≤ y` real: `ℝ` int_seg: `{i..j-}` uimplies: `b supposing a` 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` uimplies: `b supposing a` rsum: `Σ{x[k] | n≤k≤m}` 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` real: `ℝ` prop: `ℙ` int_seg: `{i..j-}` lelt: `i ≤ j < k` callbyvalueall: callbyvalueall has-value: `(a)↓` has-valueall: `has-valueall(a)` cand: `A c∧ B` top: `Top` nat: `ℕ` less_than: `a < b` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` btrue: `tt` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` bfalse: `ff` pointwise-rleq: `x[k] ≤ y[k] for k ∈ [n,m]` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]`
Lemmas referenced :  int_term_value_subtract_lemma int_formula_prop_less_lemma itermSubtract_wf intformless_wf int_formula_prop_wf int_term_value_constant_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermConstant_wf itermAdd_wf itermVar_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le subtract_wf int_seg_properties select-from-upto assert_of_bnot iff_weakening_uiff iff_transitivity eqff_to_assert assert_of_lt_int eqtt_to_assert bool_subtype_base bool_wf subtype_base_sq bool_cases not_wf bnot_wf assert_wf lt_int_wf select-map lelt_wf top_wf subtype_rel_list length-from-upto length-map length_wf nat_wf length_wf_nat map-length radd-list_functionality_wrt_rleq valueall-type-real-list evalall-reduce from-upto_wf less_than_wf le_wf and_wf map_wf real-valueall-type list-valueall-type list_wf valueall-type-has-valueall int-value-type value-type-has-value pointwise-rleq_wf nat_plus_wf real_wf int_seg_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 addEquality natural_numberEquality hypothesis setElimination rename minusEquality axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality functionEquality intEquality voidElimination independent_isectElimination setEquality callbyvalueReduce voidEquality independent_pairFormation lambdaFormation dependent_set_memberEquality productEquality unionElimination instantiate cumulativity independent_functionElimination impliesFunctionality dependent_pairFormation int_eqEquality computeAll

Latex:
\mforall{}[n,m:\mBbbZ{}].  \mforall{}[x,y:\{n..m  +  1\msupminus{}\}  {}\mrightarrow{}  \mBbbR{}].
\mSigma{}\{x[k]  |  n\mleq{}k\mleq{}m\}  \mleq{}  \mSigma{}\{y[k]  |  n\mleq{}k\mleq{}m\}  supposing  x[k]  \mleq{}  y[k]  for  k  \mmember{}  [n,m]

Date html generated: 2016_05_18-AM-07_45_11
Last ObjectModification: 2016_01_17-AM-02_07_50

Theory : reals

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