Nuprl Lemma : Riemann-sums-converge
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀f:[a, b] ⟶ℝ. ∀mc:f[x] continuous for x ∈ [a, b]. Riemann-sum(f;a;b;k + 1)↓ as k→∞
Proof
Definitions occuring in Statement :
Riemann-sum: Riemann-sum(f;a;b;k),
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
rccint: [l, u],
converges: x[n]↓ as n→∞,
rleq: x ≤ y,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
so_apply: x[s],
cauchy: cauchy(n.x[n]),
label: ...$L... t,
rfun: I ⟶ℝ,
i-finite: i-finite(I),
rccint: [l, u],
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
exists: ∃x:A. B[x],
sq_stable: SqStable(P),
squash: ↓T,
rneq: x ≠ y,
guard: {T},
satisfiable_int_formula: satisfiable_int_formula(fmla),
rless: x < y,
sq_exists: ∃x:{A| B[x]},
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
ge: i ≥ j ,
Riemann-sum: Riemann-sum(f;a;b;k),
let: let,
real: ℝ,
cand: A c∧ B,
rleq: x ≤ y,
rnonneg: rnonneg(x),
i-length: |I|,
rsub: x - y
Lemmas referenced :
converges-iff-cauchy,
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,
nat_plus_wf,
continuous_wf,
rccint_wf,
i-member_wf,
real_wf,
rfun_wf,
set_wf,
rleq_wf,
r-archimedean-implies,
i-length_wf,
general-partition-sum,
rccint-icompact,
sq_stable__rleq,
rdiv_wf,
rless-int,
nat_plus_properties,
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,
rless_wf,
int-to-real_wf,
rless-int-fractions2,
itermMultiply_wf,
int_term_value_mul_lemma,
small-reciprocal-real,
equal_wf,
rmul_wf,
rmul_preserves_rleq,
req_wf,
req_weakening,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
uiff_transitivity,
rleq_functionality,
rmul-rdiv-cancel2,
req_functionality,
req_inversion,
rmul-assoc,
rmul_functionality,
rmul_comm,
rmul-ac,
rmul-rdiv-cancel,
rmul-one-both,
nat_plus_subtype_nat,
le_wf,
all_wf,
rabs_wf,
rsub_wf,
nat_properties,
less-iff-le,
add-swap,
default-partition-choice_wf,
full-partition_wf,
uniform-partition_wf,
partition-mesh_wf,
itermAdd_wf,
intformle_wf,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
mesh-uniform-partition,
rleq-int,
sq_stable__less_than,
decidable__le,
rless_transitivity1,
rmul_preserves_rleq2,
less_than'_wf,
right_endpoint_rccint_lemma,
left_endpoint_rccint_lemma,
radd-preserves-rleq,
radd_wf,
rminus_wf,
radd_comm,
radd_functionality,
radd-rminus-assoc,
radd-zero-both,
rleq_transitivity,
rless_transitivity2,
rleq_weakening_rless,
full-partition-non-dec
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
lambdaEquality,
isectElimination,
because_Cache,
setElimination,
rename,
dependent_set_memberEquality,
hypothesis,
hypothesisEquality,
addEquality,
natural_numberEquality,
productElimination,
unionElimination,
independent_pairFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
applyEquality,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
setEquality,
imageMemberEquality,
baseClosed,
imageElimination,
inrFormation,
dependent_pairFormation,
int_eqEquality,
computeAll,
multiplyEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberFormation,
functionEquality,
isect_memberFormation,
independent_pairEquality,
axiomEquality
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a \mleq{} b\} . \mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}. \mforall{}mc:f[x] continuous for x \mmember{} [a, b].
Riemann-sum(f;a;b;k + 1)\mdownarrow{} as k\mrightarrow{}\minfty{}
Date html generated:
2017_10_03-PM-00_53_52
Last ObjectModification:
2017_07_28-AM-08_47_32
Theory : reals_2
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