Nuprl Lemma : alt-Riemann-sums-converge-ext
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀f:[a, b] ⟶ℝ. ∀mc:f[x] continuous for x ∈ [a, b]. Riemann-sum-alt(f;a;b;k + 1)↓ as k→∞
Proof
Definitions occuring in Statement :
Riemann-sum-alt: Riemann-sum-alt(f;a;b;k),
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
rccint: [l, u],
converges: x[n]↓ as n→∞,
rleq: x ≤ y,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
ifthenelse: if b then t else f fi ,
subtract: n - m,
sq_stable__rless,
sq_stable__and,
rless-cases,
integer-bound,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
squash: ↓T,
or: P ∨ Q,
guard: {T},
prop: ℙ,
has-value: (a)↓,
implies: P ⇒ Q,
all: ∀x:A. B[x],
and: P ∧ Q,
strict4: strict4(F),
uimplies: b supposing a,
top: Top,
so_apply: x[s1;s2;s3;s4],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
uall: ∀[x:A]. B[x],
Riemann-sums-cauchy,
alt-Riemann-sums-cauchy,
so_lambda: λ2x.t[x],
so_apply: x[s],
converges-iff-cauchy,
alt-Riemann-sums-converge,
member: t ∈ T
Lemmas referenced :
sq_stable__rless,
sq_stable__and,
rless-cases,
integer-bound,
Riemann-sums-cauchy,
alt-Riemann-sums-cauchy,
converges-iff-cauchy,
alt-Riemann-sums-converge
Rules used in proof :
inlFormation,
imageElimination,
imageMemberEquality,
intEquality,
inrFormation,
exceptionSqequal,
addExceptionCases,
equalitySymmetry,
equalityTransitivity,
because_Cache,
productElimination,
hypothesisEquality,
closedConclusion,
baseApply,
callbyvalueAdd,
lambdaFormation,
independent_pairFormation,
independent_isectElimination,
voidEquality,
voidElimination,
isect_memberEquality,
baseClosed,
isectElimination,
sqequalHypSubstitution,
thin,
sqequalRule,
hypothesis,
extract_by_obid,
instantiate,
cut,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
introduction
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a \mleq{} b\} . \mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}. \mforall{}mc:f[x] continuous for x \mmember{} [a, b].
Riemann-sum-alt(f;a;b;k + 1)\mdownarrow{} as k\mrightarrow{}\minfty{}
Date html generated:
2016_07_08-PM-06_00_05
Last ObjectModification:
2016_07_05-PM-03_14_51
Theory : reals
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