### Nuprl Lemma : Riemann-sum-refinement

`∀a,b:ℝ.`
`  ((a < b)`
`  `` (∀f:[a, b] ⟶ℝ. ∀mc:f[x] continuous for x ∈ [a, b]. ∀k,n:ℕ+.`
`        ((partition-mesh([a, b];uniform-partition([a, b];k)) ≤ (mc 1 n))`
`        `` (∀m:ℕ+. (|Riemann-sum(f;a;b;k) - Riemann-sum(f;a;b;m * k)| ≤ ((r1/r(n)) * (b - a)))))))`

Proof

Definitions occuring in Statement :  Riemann-sum: `Riemann-sum(f;a;b;k)` continuous: `f[x] continuous for x ∈ I` uniform-partition: `uniform-partition(I;k)` partition-mesh: `partition-mesh(I;p)` rfun: `I ⟶ℝ` rccint: `[l, u]` rdiv: `(x/y)` rleq: `x ≤ y` rless: `x < y` rabs: `|x|` rsub: `x - y` rmul: `a * b` int-to-real: `r(n)` real: `ℝ` nat_plus: `ℕ+` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q` apply: `f a` multiply: `n * m` natural_number: `\$n`
Definitions unfolded in proof :  label: `...\$L... t` not: `¬A` false: `False` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` decidable: `Dec(P)` sq_exists: `∃x:{A| B[x]}` rless: `x < y` rev_implies: `P `` Q` or: `P ∨ Q` rneq: `x ≠ y` i-member: `r ∈ I` top: `Top` rfun: `I ⟶ℝ` so_apply: `x[s]` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` true: `True` less_than': `less_than'(a;b)` squash: `↓T` less_than: `a < b` nat_plus: `ℕ+` continuous: `f[x] continuous for x ∈ I` prop: `ℙ` uimplies: `b supposing a` guard: `{T}` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` member: `t ∈ T` cand: `A c∧ B` and: `P ∧ Q` rccint: `[l, u]` i-approx: `i-approx(I;n)` implies: `P `` Q` all: `∀x:A. B[x]` has-valueall: `has-valueall(a)` callbyvalueall: callbyvalueall has-value: `(a)↓` Riemann-sum: `Riemann-sum(f;a;b;k)` i-length: `|I|`
Lemmas referenced :  right_endpoint_rccint_lemma left_endpoint_rccint_lemma value-type-has-value set-value-type int-value-type list_wf valueall-type-has-valueall list-valueall-type real-valueall-type evalall-reduce valueall-type-real-list full-partition-non-dec mul_nat_plus default-partition-choice_wf full-partition_wf uniform-partition-refines uniform-partition-increasing rccint-icompact rleq_weakening_rless partition-refinement-sum rccint_wf uniform-partition_wf nat_plus_wf rleq_wf partition-mesh_wf less_than_wf icompact_wf i-approx_wf all_wf sq_exists_wf real_wf rless_wf int-to-real_wf i-member_wf rabs_wf rsub_wf member_rccint_lemma and_wf rdiv_wf rless-int nat_plus_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf continuous_wf subtype_rel_self rfun_wf
Rules used in proof :  setEquality computeAll intEquality int_eqEquality dependent_pairFormation unionElimination inrFormation rename setElimination voidEquality voidElimination isect_memberEquality functionEquality productEquality lambdaEquality baseClosed imageMemberEquality introduction natural_numberEquality dependent_set_memberEquality applyEquality because_Cache independent_pairFormation independent_isectElimination isectElimination hypothesis independent_functionElimination productElimination hypothesisEquality thin dependent_functionElimination sqequalHypSubstitution lemma_by_obid sqequalRule cut lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution multiplyEquality equalitySymmetry equalityTransitivity equalityEquality callbyvalueReduce

Latex:
\mforall{}a,b:\mBbbR{}.
((a  <  b)
{}\mRightarrow{}  (\mforall{}f:[a,  b]  {}\mrightarrow{}\mBbbR{}.  \mforall{}mc:f[x]  continuous  for  x  \mmember{}  [a,  b].  \mforall{}k,n:\mBbbN{}\msupplus{}.
((partition-mesh([a,  b];uniform-partition([a,  b];k))  \mleq{}  (mc  1  n))
{}\mRightarrow{}  (\mforall{}m:\mBbbN{}\msupplus{}.  (|Riemann-sum(f;a;b;k)  -  Riemann-sum(f;a;b;m  *  k)|  \mleq{}  ((r1/r(n))  *  (b  -  a)))))))

Date html generated: 2016_05_18-AM-10_40_56
Last ObjectModification: 2016_01_17-AM-00_21_13

Theory : reals

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