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
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