Nuprl Lemma : rabs-Riemann-sum
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f:[a, b] ⟶ℝ]. ∀[k:ℕ+]. (|Riemann-sum(f;a;b;k)| ≤ Riemann-sum(λx.|f x|;a;b;k))
Proof
Definitions occuring in Statement :
Riemann-sum: Riemann-sum(f;a;b;k),
rfun: I ⟶ℝ,
rccint: [l, u],
rleq: x ≤ y,
rabs: |x|,
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
rfun: I ⟶ℝ,
sq_stable: SqStable(P),
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
Riemann-sum: Riemann-sum(f;a;b;k),
let: let,
uimplies: b supposing a,
squash: ↓T,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
sq_stable__rleq,
rabs_wf,
Riemann-sum_wf,
rleq_wf,
i-member_wf,
rccint_wf,
real_wf,
rccint-icompact,
rabs-partition-sum,
uniform-partition_wf,
default-partition-choice_wf,
full-partition_wf,
full-partition-non-dec,
less_than'_wf,
rsub_wf,
nat_plus_wf,
rfun_wf,
set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
dependent_set_memberEquality,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
setEquality,
independent_functionElimination,
dependent_functionElimination,
productElimination,
independent_isectElimination,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairEquality,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[b:\{b:\mBbbR{}| a \mleq{} b\} ]. \mforall{}[f:[a, b] {}\mrightarrow{}\mBbbR{}]. \mforall{}[k:\mBbbN{}\msupplus{}].
(|Riemann-sum(f;a;b;k)| \mleq{} Riemann-sum(\mlambda{}x.|f x|;a;b;k))
Date html generated:
2016_10_26-PM-00_02_38
Last ObjectModification:
2016_09_12-PM-05_37_59
Theory : reals_2
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