Nuprl Lemma : partition-sums-converge
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀f:{f:[a, b] ⟶ℝ| ifun(f;[a, b])} . ∀e:{e:ℝ| r0 < e} .
∃d:{d:ℝ| r0 < d}
∀p:partition([a, b])
((partition-mesh([a, b];p) ≤ d)
⇒ (∀y:partition-choice(full-partition([a, b];p))
(|S(λx.f[x];full-partition([a, b];p)) - ∫ f[x] dx on [a, b]| ≤ e)))
Proof
Definitions occuring in Statement :
Riemann-integral: ∫ f[x] dx on [a, b],
ifun: ifun(f;I),
partition-sum: S(f;p),
partition-choice: partition-choice(p),
partition-mesh: partition-mesh(I;p),
full-partition: full-partition(I;p),
partition: partition(I),
rfun: I ⟶ℝ,
rccint: [l, u],
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
so_apply: x[s],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
sq_stable: SqStable(P),
squash: ↓T,
rfun: I ⟶ℝ,
prop: ℙ,
subtype_rel: A ⊆r B,
ifun: ifun(f;I),
top: Top,
real-fun: real-fun(f;a;b),
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
so_lambda: λ2x.t[x],
exists: ∃x:A. B[x],
converges-to: lim n→∞.x[n] = y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
rless: x < y,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
rge: x ≥ y,
rgt: x > y,
i-finite: i-finite(I),
rccint: [l, u],
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
Riemann-sum: Riemann-sum(f;a;b;k),
let: let,
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
le: A ≤ B,
i-length: |I|,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
less_than': less_than'(a;b),
less_than: a < b
Lemmas referenced :
rccint-icompact,
sq_stable__rleq,
general-partition-sum-no-mc,
rccint_wf,
real_wf,
i-member_wf,
subtype_rel_self,
rfun_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
req_functionality,
req_weakening,
req_wf,
set_wf,
ifun_wf,
Riemann-sums-converge-to,
partition-choice_wf,
full-partition_wf,
rleq_wf,
partition-mesh_wf,
partition_wf,
all_wf,
rabs_wf,
rsub_wf,
partition-sum_wf,
Riemann-integral_wf,
rless_wf,
int-to-real_wf,
rleq-iff-all-rless,
small-reciprocal-real,
radd_wf,
rdiv_wf,
rless-int,
nat_properties,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
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,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening_rless,
radd_functionality_wrt_rless1,
r-archimedean-implies2,
i-length_wf,
imax_wf,
nat_plus_subtype_nat,
nat_wf,
decidable__le,
intformle_wf,
intformeq_wf,
int_formula_prop_le_lemma,
int_formula_prop_eq_lemma,
equal_wf,
le_wf,
imax_ub,
itermAdd_wf,
int_term_value_add_lemma,
rleq-int,
ifthenelse_wf,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
squash_wf,
true_wf,
add_functionality_wrt_eq,
imax_unfold,
iff_weakening_equal,
rmul_preserves_rleq2,
icompact-length-nonneg,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
rleq-int-fractions,
less_than_wf,
add-is-int-iff,
int_term_value_mul_lemma,
false_wf,
rleq-implies-rleq,
rleq_functionality,
rleq_transitivity,
req-iff-rsub-is-0,
rinv_wf2,
req_transitivity,
rmul_functionality,
req_inversion,
radd-int,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma,
uniform-partition_wf,
mesh-uniform-partition,
default-partition-choice_wf,
full-partition-non-dec,
radd_functionality_wrt_rleq,
r-triangle-inequality2,
nat_plus_wf,
add_nat_plus,
Riemann-sum_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
productElimination,
independent_functionElimination,
setElimination,
rename,
isectElimination,
hypothesis,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality,
lambdaEquality,
applyEquality,
setEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
dependent_pairFormation,
functionEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
inrFormation,
unionElimination,
approximateComputation,
int_eqEquality,
intEquality,
independent_pairFormation,
applyLambdaEquality,
inlFormation,
addEquality,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
universeEquality,
multiplyEquality,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a \mleq{} b\} . \mforall{}f:\{f:[a, b] {}\mrightarrow{}\mBbbR{}| ifun(f;[a, b])\} . \mforall{}e:\{e:\mBbbR{}| r0 < e\} .
\mexists{}d:\{d:\mBbbR{}| r0 < d\}
\mforall{}p:partition([a, b])
((partition-mesh([a, b];p) \mleq{} d)
{}\mRightarrow{} (\mforall{}y:partition-choice(full-partition([a, b];p))
(|S(\mlambda{}x.f[x];full-partition([a, b];p)) - \mint{} f[x] dx on [a, b]| \mleq{} e)))
Date html generated:
2019_10_30-AM-11_38_40
Last ObjectModification:
2018_08_23-PM-02_22_04
Theory : reals_2
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