Nuprl Lemma : nearby-partition-sum-ext
∀I:Interval
(icompact(I)
⇒ iproper(I)
⇒ (∀f:I ⟶ℝ. ∀mc:f[x] continuous for x ∈ I. ∀p:partition(I). ∀x:partition-choice(full-partition(I;p)).
∀alpha:{a:ℝ| r0 < a} .
∃e:{e:ℝ| r0 < e}
∀q:partition(I). ∀y:partition-choice(full-partition(I;q)).
(nearby-partitions(e;p;q)
⇒ (∀i:ℕ||p|| + 1. (|x[i] - y[i]| ≤ e))
⇒ (|S(f;full-partition(I;q)) - S(f;full-partition(I;p))| ≤ alpha))))
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
partition-sum: S(f;p),
partition-choice-ap: x[i],
partition-choice: partition-choice(p),
full-partition: full-partition(I;p),
nearby-partitions: nearby-partitions(e;p;q),
partition: partition(I),
icompact: icompact(I),
rfun: I ⟶ℝ,
iproper: iproper(I),
interval: Interval,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
length: ||as||,
int_seg: {i..j-},
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
i-length: |I|,
so_apply: x[s],
nearby-partition-sum,
small-reciprocal-real-ext,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
uimplies: b supposing a,
so_lambda: λ2x.t[x]
Lemmas referenced :
nearby-partition-sum,
lifting-strict-callbyvalue,
istype-void,
strict4-spread,
small-reciprocal-real-ext
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination
Latex:
\mforall{}I:Interval
(icompact(I)
{}\mRightarrow{} iproper(I)
{}\mRightarrow{} (\mforall{}f:I {}\mrightarrow{}\mBbbR{}. \mforall{}mc:f[x] continuous for x \mmember{} I. \mforall{}p:partition(I).
\mforall{}x:partition-choice(full-partition(I;p)). \mforall{}alpha:\{a:\mBbbR{}| r0 < a\} .
\mexists{}e:\{e:\mBbbR{}| r0 < e\}
\mforall{}q:partition(I). \mforall{}y:partition-choice(full-partition(I;q)).
(nearby-partitions(e;p;q)
{}\mRightarrow{} (\mforall{}i:\mBbbN{}||p|| + 1. (|x[i] - y[i]| \mleq{} e))
{}\mRightarrow{} (|S(f;full-partition(I;q)) - S(f;full-partition(I;p))| \mleq{} alpha))))
Date html generated:
2019_10_30-AM-11_37_12
Last ObjectModification:
2019_01_27-PM-04_38_16
Theory : reals_2
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