Nuprl Lemma : nearby-separated-partition-sum
∀I:Interval
(icompact(I)
⇒ iproper(I)
⇒ (∀f:I ⟶ℝ. ∀mc:f[x] continuous for x ∈ I. ∀alpha,e:{e:ℝ| r0 < e} . ∀p,q:partition(I).
∀x:partition-choice(full-partition(I;p)). ∀y:partition-choice(full-partition(I;q)).
∃p':partition(I)
∃x':partition-choice(full-partition(I;p'))
((partition-mesh(I;p') ≤ (partition-mesh(I;p) + e))
∧ (∃q':partition(I)
∃y':partition-choice(full-partition(I;q'))
(separated-partitions(p';q')
∧ (partition-mesh(I;q') ≤ (partition-mesh(I;q) + e))
∧ (|S(f;full-partition(I;p)) - S(f;full-partition(I;q))| ≤ (|S(f;full-partition(I;p'))
- S(f;full-partition(I;q'))|
+ alpha)))))))
Proof
Definitions occuring in Statement :
separated-partitions: separated-partitions(P;Q),
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,
iproper: iproper(I),
interval: Interval,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
radd: 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,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
label: ...$L... t,
rfun: I ⟶ℝ,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
cand: A c∧ B,
rge: x ≥ y,
partition: partition(I),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rmul_preserves_rless,
rdiv_wf,
rless-int,
nearby-partition-sum,
int-to-real_wf,
rless_wf,
partition-choice_wf,
full-partition_wf,
partition_wf,
set_wf,
real_wf,
continuous_wf,
i-member_wf,
rfun_wf,
iproper_wf,
icompact_wf,
interval_wf,
rmul_wf,
rmul-zero-both,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
sq_stable__rless,
rless_functionality,
req_transitivity,
rmul-rinv3,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rmin_strict_ub,
rmin_wf,
nearby-separated-partitions,
exists_wf,
rleq_wf,
partition-mesh_wf,
radd_wf,
separated-partitions_wf,
rabs_wf,
rsub_wf,
partition-sum_wf,
nearby-partitions_functionality,
rmin-rleq,
rmin_lb,
nearby-partition-choice,
rleq_weakening_equal,
nearby-partition-mesh,
rleq_functionality,
rabs-difference-symmetry,
req_weakening,
equal_wf,
req_functionality,
radd_functionality,
uimplies_transitivity,
rleq_functionality_wrt_implies,
r-triangle-inequality2,
radd_functionality_wrt_rleq,
rleq_weakening,
itermAdd_wf,
real_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
isectElimination,
independent_isectElimination,
sqequalRule,
hypothesis,
inrFormation,
productElimination,
independent_functionElimination,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
setElimination,
rename,
dependent_set_memberEquality,
lambdaEquality,
applyEquality,
setEquality,
imageElimination,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
addLevel,
levelHypothesis,
andLevelFunctionality,
productEquality,
dependent_pairFormation,
inlFormation,
equalityTransitivity,
equalitySymmetry
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{}alpha,e:\{e:\mBbbR{}| r0 < e\} . \mforall{}p,q:partition(I).
\mforall{}x:partition-choice(full-partition(I;p)). \mforall{}y:partition-choice(full-partition(I;q)).
\mexists{}p':partition(I)
\mexists{}x':partition-choice(full-partition(I;p'))
((partition-mesh(I;p') \mleq{} (partition-mesh(I;p) + e))
\mwedge{} (\mexists{}q':partition(I)
\mexists{}y':partition-choice(full-partition(I;q'))
(separated-partitions(p';q')
\mwedge{} (partition-mesh(I;q') \mleq{} (partition-mesh(I;q) + e))
\mwedge{} (|S(f;full-partition(I;p))
- S(f;full-partition(I;q))| \mleq{} (|S(f;full-partition(I;p'))
- S(f;full-partition(I;q'))|
+ alpha)))))))
Date html generated:
2019_10_30-AM-11_37_26
Last ObjectModification:
2018_08_23-PM-00_17_00
Theory : reals_2
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