Nuprl Lemma : Riemann-sums-near

`∀a,b:ℝ.`
`  ((a < b)`
`  `` (∀f:[a, b] ⟶ℝ. ∀mc:f[x] continuous for x ∈ [a, b]. ∀k,m,n:ℕ+.`
`        (((b - a/r(k)) ≤ (mc 1 n))`
`        `` ((b - a/r(m)) ≤ (mc 1 n))`
`        `` (|Riemann-sum(f;a;b;k) - Riemann-sum(f;a;b;m)| ≤ ((r(2)/r(n)) * (b - a))))))`

Proof

Definitions occuring in Statement :  Riemann-sum: `Riemann-sum(f;a;b;k)` continuous: `f[x] continuous for x ∈ I` 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` natural_number: `\$n`
Definitions unfolded in proof :  label: `...\$L... t` i-member: `r ∈ I` 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` continuous: `f[x] continuous for x ∈ I` top: `Top` 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` nat_plus: `ℕ+` prop: `ℙ` uimplies: `b supposing a` guard: `{T}` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` cand: `A c∧ B` and: `P ∧ Q` rccint: `[l, u]` i-approx: `i-approx(I;n)` implies: `P `` Q` member: `t ∈ T` all: `∀x:A. B[x]` i-length: `|I|` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` icompact: `icompact(I)` rge: `x ≥ y`
Lemmas referenced :  rmul-rdiv-cancel rmul-ac rmul_comm rmul-assoc req_functionality rmul-int-rdiv radd-int rmul-identity1 radd_functionality req_transitivity rmul-distrib2 req_inversion rmul_functionality uiff_transitivity rmul_preserves_rleq req_wf rleq_functionality_wrt_implies Riemann-sum_wf radd_wf mul_nat_plus rleq_weakening_equal r-triangle-inequality2 rmul_wf rabs-difference-symmetry radd_functionality_wrt_rleq mul-commutes req_weakening mesh-uniform-partition rleq_functionality partition-mesh_wf uniform-partition_wf i-length_wf right_endpoint_rccint_lemma left_endpoint_rccint_lemma Riemann-sum-refinement rccint-icompact rleq_weakening_rless rleq_wf rdiv_wf rsub_wf int-to-real_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 rless_wf less_than_wf icompact_wf i-approx_wf rccint_wf all_wf nat_plus_wf sq_exists_wf real_wf i-member_wf rabs_wf member_rccint_lemma and_wf continuous_wf subtype_rel_self rfun_wf
Rules used in proof :  setEquality functionEquality productEquality baseClosed imageMemberEquality introduction dependent_set_memberEquality applyEquality computeAll voidEquality voidElimination isect_memberEquality intEquality int_eqEquality lambdaEquality dependent_pairFormation unionElimination natural_numberEquality because_Cache inrFormation rename setElimination independent_pairFormation independent_isectElimination isectElimination productElimination sqequalRule independent_functionElimination hypothesisEquality thin dependent_functionElimination sqequalHypSubstitution hypothesis lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution lemma_by_obid cut equalitySymmetry equalityTransitivity equalityEquality multiplyEquality addEquality

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,m,n:\mBbbN{}\msupplus{}.
(((b  -  a/r(k))  \mleq{}  (mc  1  n))
{}\mRightarrow{}  ((b  -  a/r(m))  \mleq{}  (mc  1  n))
{}\mRightarrow{}  (|Riemann-sum(f;a;b;k)  -  Riemann-sum(f;a;b;m)|  \mleq{}  ((r(2)/r(n))  *  (b  -  a))))))

Date html generated: 2016_05_18-AM-10_41_15
Last ObjectModification: 2016_01_17-AM-00_24_26

Theory : reals

Home Index