Nuprl Lemma : Riemann-sum-refinement
∀a,b:ℝ.
((a < b)
⇒ (∀f:[a, b] ⟶ℝ. ∀mc:f[x] continuous for x ∈ [a, b]. ∀k,n:ℕ+.
((partition-mesh([a, b];uniform-partition([a, b];k)) ≤ (mc 1 n))
⇒ (∀m:ℕ+. (|Riemann-sum(f;a;b;k) - Riemann-sum(f;a;b;m * k)| ≤ ((r1/r(n)) * (b - a)))))))
Proof
Definitions occuring in Statement :
Riemann-sum: Riemann-sum(f;a;b;k)
,
continuous: f[x] continuous for x ∈ I
,
uniform-partition: uniform-partition(I;k)
,
partition-mesh: partition-mesh(I;p)
,
rfun: I ⟶ℝ
,
rccint: [l, u]
,
rdiv: (x/y)
,
rleq: x ≤ y
,
rless: x < y
,
rabs: |x|
,
rsub: x - y
,
rmul: a * b
,
int-to-real: r(n)
,
real: ℝ
,
nat_plus: ℕ+
,
so_apply: x[s]
,
all: ∀x:A. B[x]
,
implies: P
⇒ Q
,
apply: f a
,
multiply: n * m
,
natural_number: $n
Definitions unfolded in proof :
label: ...$L... t
,
not: ¬A
,
false: False
,
exists: ∃x:A. B[x]
,
satisfiable_int_formula: satisfiable_int_formula(fmla)
,
decidable: Dec(P)
,
sq_exists: ∃x:{A| B[x]}
,
rless: x < y
,
rev_implies: P
⇐ Q
,
or: P ∨ Q
,
rneq: x ≠ y
,
i-member: r ∈ I
,
top: Top
,
rfun: I ⟶ℝ
,
so_apply: x[s]
,
so_lambda: λ2x.t[x]
,
subtype_rel: A ⊆r B
,
true: True
,
less_than': less_than'(a;b)
,
squash: ↓T
,
less_than: a < b
,
nat_plus: ℕ+
,
continuous: f[x] continuous for x ∈ I
,
prop: ℙ
,
uimplies: b supposing a
,
guard: {T}
,
uall: ∀[x:A]. B[x]
,
iff: P
⇐⇒ Q
,
member: t ∈ T
,
cand: A c∧ B
,
and: P ∧ Q
,
rccint: [l, u]
,
i-approx: i-approx(I;n)
,
implies: P
⇒ Q
,
all: ∀x:A. B[x]
,
has-valueall: has-valueall(a)
,
callbyvalueall: callbyvalueall,
has-value: (a)↓
,
Riemann-sum: Riemann-sum(f;a;b;k)
,
i-length: |I|
Lemmas referenced :
right_endpoint_rccint_lemma,
left_endpoint_rccint_lemma,
value-type-has-value,
set-value-type,
int-value-type,
list_wf,
valueall-type-has-valueall,
list-valueall-type,
real-valueall-type,
evalall-reduce,
valueall-type-real-list,
full-partition-non-dec,
mul_nat_plus,
default-partition-choice_wf,
full-partition_wf,
uniform-partition-refines,
uniform-partition-increasing,
rccint-icompact,
rleq_weakening_rless,
partition-refinement-sum,
rccint_wf,
uniform-partition_wf,
nat_plus_wf,
rleq_wf,
partition-mesh_wf,
less_than_wf,
icompact_wf,
i-approx_wf,
all_wf,
sq_exists_wf,
real_wf,
rless_wf,
int-to-real_wf,
i-member_wf,
rabs_wf,
rsub_wf,
member_rccint_lemma,
and_wf,
rdiv_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,
continuous_wf,
subtype_rel_self,
rfun_wf
Rules used in proof :
setEquality,
computeAll,
intEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
inrFormation,
rename,
setElimination,
voidEquality,
voidElimination,
isect_memberEquality,
functionEquality,
productEquality,
lambdaEquality,
baseClosed,
imageMemberEquality,
introduction,
natural_numberEquality,
dependent_set_memberEquality,
applyEquality,
because_Cache,
independent_pairFormation,
independent_isectElimination,
isectElimination,
hypothesis,
independent_functionElimination,
productElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
lemma_by_obid,
sqequalRule,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
multiplyEquality,
equalitySymmetry,
equalityTransitivity,
equalityEquality,
callbyvalueReduce
Latex:
\mforall{}a,b:\mBbbR{}.
((a < b)
{}\mRightarrow{} (\mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}. \mforall{}mc:f[x] continuous for x \mmember{} [a, b]. \mforall{}k,n:\mBbbN{}\msupplus{}.
((partition-mesh([a, b];uniform-partition([a, b];k)) \mleq{} (mc 1 n))
{}\mRightarrow{} (\mforall{}m:\mBbbN{}\msupplus{}. (|Riemann-sum(f;a;b;k) - Riemann-sum(f;a;b;m * k)| \mleq{} ((r1/r(n)) * (b - a)))))))
Date html generated:
2016_05_18-AM-10_40_56
Last ObjectModification:
2016_01_17-AM-00_21_13
Theory : reals
Home
Index