Nuprl Lemma : Riemann-sum-rleq
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f,g:[a, b] ⟶ℝ]. ∀[k:ℕ+].
Riemann-sum(f;a;b;k) ≤ Riemann-sum(g;a;b;k) supposing ∀x:ℝ. ((x ∈ [a, b]) ⇒ ((f x) ≤ (g x)))
Proof
Definitions occuring in Statement :
Riemann-sum: Riemann-sum(f;a;b;k),
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
real: ℝ,
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
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,
squash: ↓T,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s]
Lemmas referenced :
sq_stable__rleq,
Riemann-sum_wf,
rleq_wf,
rccint-icompact,
partition-sum-rleq,
rccint_wf,
uniform-partition_wf,
default-partition-choice_wf,
full-partition_wf,
full-partition-non-dec,
less_than'_wf,
rsub_wf,
real_wf,
nat_plus_wf,
all_wf,
i-member_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,
independent_functionElimination,
dependent_functionElimination,
productElimination,
sqequalRule,
independent_isectElimination,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality,
independent_pairEquality,
applyEquality,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
isect_memberEquality,
voidElimination
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[b:\{b:\mBbbR{}| a \mleq{} b\} ]. \mforall{}[f,g:[a, b] {}\mrightarrow{}\mBbbR{}]. \mforall{}[k:\mBbbN{}\msupplus{}].
Riemann-sum(f;a;b;k) \mleq{} Riemann-sum(g;a;b;k) supposing \mforall{}x:\mBbbR{}. ((x \mmember{} [a, b]) {}\mRightarrow{} ((f x) \mleq{} (g x)))
Date html generated:
2016_10_26-PM-00_02_35
Last ObjectModification:
2016_09_12-PM-05_37_57
Theory : reals_2
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