Nuprl Lemma : general-partition-sum-no-mc
∀I:Interval
(icompact(I)
⇒ (∀f:{f:I ⟶ℝ| ifun(f;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)))
Proof
Definitions occuring in Statement :
ifun: ifun(f;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,
interval: Interval,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
ifun: ifun(f;I),
top: Top,
subtype_rel: A ⊆r B,
icompact: icompact(I),
and: P ∧ Q,
exists: ∃x:A. B[x],
rfun: I ⟶ℝ
Lemmas referenced :
real_wf,
rless_wf,
int-to-real_wf,
rfun_wf,
ifun_wf,
icompact_wf,
interval_wf,
icompact-is-rccint,
sq_stable__ifun,
left_endpoint_rccint_lemma,
istype-void,
right_endpoint_rccint_lemma,
real-fun-bounded,
left-endpoint_wf,
subtype_rel_self,
rccint_wf,
right-endpoint_wf,
icompact-endpoints-rleq,
rleq_wf,
general-partition-sum-from-bound,
i-member_wf,
rabs_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
setIsType,
universeIsType,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesisEquality,
independent_isectElimination,
sqequalRule,
setElimination,
rename,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
applyEquality,
because_Cache,
productElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
functionIsType,
inhabitedIsType
Latex:
\mforall{}I:Interval
(icompact(I)
{}\mRightarrow{} (\mforall{}f:\{f:I {}\mrightarrow{}\mBbbR{}| ifun(f;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)))
Date html generated:
2019_10_30-AM-11_38_26
Last ObjectModification:
2019_01_27-PM-05_15_55
Theory : reals_2
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