Nuprl Lemma : separated-partition-sum
∀I:Interval
(icompact(I)
⇒ (∀f:I ⟶ℝ. ∀mc:f[x] continuous for x ∈ I. ∀q:partition(I). ∀n:ℕ+.
((partition-mesh(I;q) ≤ (mc 1 n))
⇒ (∀p:partition(I). ∀m:ℕ+.
((partition-mesh(I;p) ≤ (mc 1 m))
⇒ separated-partitions(p;q)
⇒ (∀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))| ≤ (((r1/r(n)) + (r1/r(m))) * |I|))))))))
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,
i-length: |I|,
interval: Interval,
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
nat_plus: ℕ+,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
continuous: f[x] continuous for x ∈ I,
uall: ∀[x:A]. B[x],
prop: ℙ,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
rfun: I ⟶ℝ,
nat_plus: ℕ+,
rneq: x ≠ y,
or: P ∨ Q,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
partition-refines: P refines Q,
less_than: a < b,
less_than': less_than'(a;b),
label: ...$L... t,
separated-partitions: separated-partitions(P;Q),
icompact: icompact(I)
Lemmas referenced :
nat_plus_wf,
icompact_wf,
i-approx_wf,
squash_wf,
true_wf,
i-approx-of-compact,
iff_weakening_equal,
all_wf,
sq_exists_wf,
i-member_wf,
rleq_wf,
rabs_wf,
rsub_wf,
real_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
subtype_rel_sets,
rleq_functionality_wrt_implies,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf,
rleq_weakening_equal,
rleq_weakening,
real_term_polynomial,
itermSubtract_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
separated-partitions-have-common-refinement,
partition-choice_wf,
full-partition_wf,
separated-partitions_wf,
partition-mesh_wf,
less_than_wf,
partition_wf,
continuous_wf,
rfun_wf,
interval_wf,
partition-refinement-sum,
default-partition-choice_wf,
full-partition-non-dec,
partition-sum_wf,
radd_wf,
r-triangle-inequality2,
rmul_wf,
i-length_wf,
radd_functionality_wrt_rleq,
rleq_functionality,
rabs-difference-symmetry,
req_weakening,
itermAdd_wf,
itermMultiply_wf,
real_term_value_add_lemma,
real_term_value_mul_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
applyEquality,
functionExtensionality,
hypothesisEquality,
setEquality,
introduction,
extract_by_obid,
hypothesis,
isectElimination,
thin,
because_Cache,
dependent_set_memberEquality,
lambdaEquality,
imageElimination,
dependent_functionElimination,
independent_functionElimination,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
productEquality,
functionEquality,
universeEquality,
setElimination,
rename,
inrFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}I:Interval
(icompact(I)
{}\mRightarrow{} (\mforall{}f:I {}\mrightarrow{}\mBbbR{}. \mforall{}mc:f[x] continuous for x \mmember{} I. \mforall{}q:partition(I). \mforall{}n:\mBbbN{}\msupplus{}.
((partition-mesh(I;q) \mleq{} (mc 1 n))
{}\mRightarrow{} (\mforall{}p:partition(I). \mforall{}m:\mBbbN{}\msupplus{}.
((partition-mesh(I;p) \mleq{} (mc 1 m))
{}\mRightarrow{} separated-partitions(p;q)
{}\mRightarrow{} (\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{} (((r1/r(n))
+ (r1/r(m)))
* |I|))))))))
Date html generated:
2017_10_03-PM-00_49_33
Last ObjectModification:
2017_07_28-AM-08_47_19
Theory : reals_2
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