Nuprl Lemma : partition-sum-bound
∀I:Interval
(icompact(I)
⇒ (∀f:I ⟶ℝ. ∀mc:f[x] continuous for x ∈ I. ∀p:partition(I). ∀y:partition-choice(full-partition(I;p)).
(|S(f;full-partition(I;p))| ≤ (||f[x]||_I * |I|))))
Proof
Definitions occuring in Statement :
Inorm: ||f[x]||_I,
continuous: f[x] continuous for x ∈ I,
partition-sum: S(f;p),
partition-choice: partition-choice(p),
full-partition: full-partition(I;p),
partition: partition(I),
icompact: icompact(I),
rfun: I ⟶ℝ,
i-length: |I|,
interval: Interval,
rleq: x ≤ y,
rabs: |x|,
rmul: a * b,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
partition-sum: S(f;p),
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
squash: ↓T,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
full-partition: full-partition(I;p),
top: Top,
partition: partition(I),
subtype_rel: A ⊆r B,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
false: False,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
icompact: icompact(I),
partition-choice: partition-choice(p),
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
uiff: uiff(P;Q),
l_all: (∀x∈L.P[x]),
i-member: r ∈ I,
rccint: [l, u],
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
frs-non-dec: frs-non-dec(L),
req_int_terms: t1 ≡ t2,
assert: ↑b,
ifthenelse: if b then t else f fi ,
null: null(as),
cons: [a / b],
bfalse: ff,
less_than': less_than'(a;b),
select: L[n],
last: last(L),
subtract: n - m,
i-length: |I|
Lemmas referenced :
partition-choice_wf,
full-partition_wf,
partition_wf,
continuous_wf,
real_wf,
i-member_wf,
rfun_wf,
icompact_wf,
interval_wf,
le_witness_for_triv,
length_of_cons_lemma,
istype-void,
length_nil,
non_neg_length,
nil_wf,
length_cons,
right-endpoint_wf,
cons_wf,
append_wf,
length_append,
subtype_rel_list,
top_wf,
length-append,
length_of_nil_lemma,
decidable__equal_int,
length_wf,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
int_seg_properties,
decidable__le,
intformand_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
decidable__lt,
subtract_wf,
add-is-int-iff,
intformless_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
false_wf,
istype-le,
istype-less_than,
full-partition-point-member,
select_wf,
i-member-between,
rccint_wf,
int_seg_wf,
rabs_wf,
rsum_wf,
rmul_wf,
rsub_wf,
Inorm_wf,
i-length_wf,
rleq_functionality_wrt_implies,
rabs-rsum,
rleq_weakening_equal,
rsum_functionality_wrt_rleq,
rleq_functionality,
rabs-rmul,
req_weakening,
req_functionality,
rabs-of-nonneg,
full-partition-non-dec,
radd-preserves-rleq,
int-to-real_wf,
radd_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
Inorm-bound,
req_wf,
rmul_preserves_rleq2,
zero-rleq-rabs,
rmul_functionality,
req_inversion,
rsum_linearity2,
last_wf,
istype-assert,
null_wf3,
istype-false,
frs-non-dec-sum,
last-cons,
null_append,
null_cons_lemma,
band-bfalse,
last_append
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
inhabitedIsType,
hypothesis,
thin,
equalityIstype,
hypothesisEquality,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
independent_isectElimination,
sqequalRule,
lambdaEquality_alt,
applyEquality,
setIsType,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
functionIsTypeImplies,
isect_memberEquality_alt,
voidElimination,
voidEquality,
because_Cache,
setElimination,
rename,
addEquality,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
functionExtensionality,
dependent_set_memberEquality_alt,
independent_pairFormation,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
productIsType
Latex:
\mforall{}I:Interval
(icompact(I)
{}\mRightarrow{} (\mforall{}f:I {}\mrightarrow{}\mBbbR{}. \mforall{}mc:f[x] continuous for x \mmember{} I. \mforall{}p:partition(I).
\mforall{}y:partition-choice(full-partition(I;p)).
(|S(f;full-partition(I;p))| \mleq{} (||f[x]||\_I * |I|))))
Date html generated:
2019_10_30-AM-11_37_46
Last ObjectModification:
2019_01_27-PM-04_50_15
Theory : reals_2
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