Nuprl Lemma : general-partition-sum-ext
∀I:Interval
(icompact(I)
⇒ (∀f:I ⟶ℝ. ∀mc:f[x] continuous for x ∈ I. ∀e:{e:ℝ| r0 < e} .
∃d:{d:ℝ| r0 < d}
∀p,q:{p:partition(I)| partition-mesh(I;p) ≤ d} . ∀x:partition-choice(full-partition(I;p)).
∀y:partition-choice(full-partition(I;q)).
(|S(f;full-partition(I;q)) - S(f;full-partition(I;p))| ≤ (e * |I|))))
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
partition-sum: S(f;p),
partition-choice: partition-choice(p),
partition-mesh: partition-mesh(I;p),
full-partition: full-partition(I;p),
partition: partition(I),
icompact: icompact(I),
rfun: I ⟶ℝ,
i-length: |I|,
interval: Interval,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
general-partition-sum,
small-reciprocal-real-ext,
sq_stable__and,
sq_stable__rless,
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],
so_apply: x[s]
Lemmas referenced :
general-partition-sum,
lifting-strict-callbyvalue,
istype-void,
strict4-spread,
small-reciprocal-real-ext,
sq_stable__and,
sq_stable__rless
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{} (\mforall{}f:I {}\mrightarrow{}\mBbbR{}. \mforall{}mc:f[x] continuous for x \mmember{} I. \mforall{}e:\{e:\mBbbR{}| r0 < e\} .
\mexists{}d:\{d:\mBbbR{}| r0 < d\}
\mforall{}p,q:\{p:partition(I)| partition-mesh(I;p) \mleq{} d\} . \mforall{}x:partition-choice(full-partition(I;p)).
\mforall{}y:partition-choice(full-partition(I;q)).
(|S(f;full-partition(I;q)) - S(f;full-partition(I;p))| \mleq{} (e * |I|))))
Date html generated:
2019_10_30-AM-11_38_16
Last ObjectModification:
2019_01_27-PM-05_14_40
Theory : reals_2
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