Nuprl Lemma : Riemann-sums-converge-to
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀f:{f:[a, b] ⟶ℝ| ifun(f;[a, b])} . lim k→∞.Riemann-sum(λx.f[x];a;b;k + 1) = ∫ f[x] dx on [a, b\000C]
Proof
Definitions occuring in Statement :
Riemann-integral: ∫ f[x] dx on [a, b],
Riemann-sum: Riemann-sum(f;a;b;k),
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
converges-to: lim n→∞.x[n] = y,
rleq: x ≤ y,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
lambda: λx.A[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
Riemann-integral: ∫ f[x] dx on [a, b],
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
nat_plus: ℕ+,
nat: ℕ,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
subtract: n - m,
less_than': less_than'(a;b),
true: True,
so_apply: x[s],
top: Top,
sq_stable: SqStable(P),
squash: ↓T,
rfun: I ⟶ℝ,
guard: {T},
converges: x[n]↓ as n→∞,
exists: ∃x:A. B[x],
pi1: fst(t)
Lemmas referenced :
Riemann-sums-converge-no-mc,
all_wf,
real_wf,
rleq_wf,
rfun_wf,
rccint_wf,
ifun_wf,
rccint-icompact,
converges_wf,
Riemann-sum_wf,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
nat_wf,
sq_stable__rleq,
equal_wf,
set_wf,
i-member_wf,
subtype_rel_self,
squash_wf,
icompact_wf,
interval_wf,
eta_conv,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
lambdaEquality,
sqequalHypSubstitution,
sqequalRule,
hypothesisEquality,
introduction,
isectElimination,
setEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
productElimination,
independent_functionElimination,
dependent_set_memberEquality,
addEquality,
natural_numberEquality,
unionElimination,
independent_pairFormation,
voidElimination,
minusEquality,
isect_memberEquality,
voidEquality,
intEquality,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
universeEquality
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a \mleq{} b\} . \mforall{}f:\{f:[a, b] {}\mrightarrow{}\mBbbR{}| ifun(f;[a, b])\} .
lim k\mrightarrow{}\minfty{}.Riemann-sum(\mlambda{}x.f[x];a;b;k + 1) = \mint{} f[x] dx on [a, b]
Date html generated:
2016_10_26-PM-00_02_08
Last ObjectModification:
2016_09_12-PM-05_37_50
Theory : reals_2
Home
Index