Nuprl Lemma : general-partition-sum-from-bound
∀I:Interval
(icompact(I)
⇒ (∀f:{f:I ⟶ℝ| ifun(f;I)} . ∀b:{b:ℝ| (r0 ≤ b) ∧ (∀x:ℝ. ((x ∈ I) ⇒ (|f x| ≤ b)))} . ∀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,
i-member: r ∈ I,
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,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
natural_number: $n
Definitions unfolded in proof :
icompact: icompact(I),
rfun: I ⟶ℝ,
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
uimplies: b supposing a,
squash: ↓T,
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rneq: x ≠ y,
rev_implies: P ⇐ Q,
rdiv: (x/y),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
exists: ∃x:A. B[x],
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
rgt: x > y,
subtype_rel: A ⊆r B,
true: True,
less_than': less_than'(a;b),
less_than: a < b
Lemmas referenced :
interval_wf,
icompact_wf,
ifun_wf,
rfun_wf,
rabs_wf,
i-member_wf,
rleq_wf,
rless_wf,
real_wf,
i-length_wf,
radd_wf,
rmul_wf,
sq_stable__rless,
int-to-real_wf,
rless-cases,
partition-sum-bound-no-mc,
rmul-is-positive,
ifun-continuous,
general-partition-sum-ext,
rdiv_wf,
rmul_preserves_rless,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
rinv_wf2,
rless_functionality,
req_transitivity,
rmul-rinv3,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
partition-choice_wf,
full-partition_wf,
partition_wf,
partition-mesh_wf,
rsub_wf,
partition-sum_wf,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rleq_functionality,
req_weakening,
rless_irreflexivity,
rless_transitivity1,
icompact-length-nonneg,
real_term_value_minus_lemma,
req_functionality,
rabs_functionality,
rleq_weakening_rless,
iff_weakening_equal,
subtype_rel_self,
rabs-rminus,
true_wf,
squash_wf,
req_wf,
itermMinus_wf,
rminus_wf,
rleq_weakening,
rless-int,
real_term_value_add_lemma,
itermAdd_wf,
radd_functionality_wrt_rleq,
r-triangle-inequality2
Rules used in proof :
productElimination,
dependent_set_memberEquality_alt,
applyEquality,
functionIsType,
productIsType,
universeIsType,
setIsType,
unionElimination,
independent_isectElimination,
because_Cache,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
natural_numberEquality,
isectElimination,
rename,
setElimination,
hypothesis,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
inrFormation_alt,
closedConclusion,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
dependent_pairFormation_alt,
equalityTransitivity,
equalitySymmetry,
universeEquality,
instantiate,
inhabitedIsType,
independent_pairFormation
Latex:
\mforall{}I:Interval
(icompact(I)
{}\mRightarrow{} (\mforall{}f:\{f:I {}\mrightarrow{}\mBbbR{}| ifun(f;I)\} . \mforall{}b:\{b:\mBbbR{}| (r0 \mleq{} b) \mwedge{} (\mforall{}x:\mBbbR{}. ((x \mmember{} I) {}\mRightarrow{} (|f x| \mleq{} b)))\} . \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_21
Last ObjectModification:
2019_10_10-AM-10_21_07
Theory : reals_2
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