Nuprl Lemma : fun-converges-to-integral
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ. ∀F:I ⟶ℝ.
(lim n→∞.f[n;x] = λy.F[y] for x ∈ I
⇒ (∀n:ℕ. ∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ (f[n;x] = f[n;y])))
⇒ (∀a:{a:ℝ| a ∈ I} . lim n→∞.a_∫-x f[n;t] dt = λx.a_∫-x F[t] dt for x ∈ I))
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
req: x = y,
real: ℝ,
nat: ℕ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
rfun: I ⟶ℝ,
so_lambda: λ2x y.t[x; y],
label: ...$L... t,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
sq_stable: SqStable(P),
squash: ↓T,
and: P ∧ Q,
cand: A c∧ B,
uimplies: b supposing a,
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subinterval: I ⊆ J ,
icompact: icompact(I),
i-nonvoid: i-nonvoid(I),
nat: ℕ,
ge: i ≥ j ,
int_upper: {i...},
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
rneq: x ≠ y,
less_than: a < b,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
uiff: uiff(P;Q),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
rless: x < y,
sq_exists: ∃x:A [B[x]],
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Lemmas referenced :
real_wf,
i-member_wf,
istype-nat,
req_wf,
subtype_rel_self,
fun-converges-to_wf,
rfun_wf,
interval_wf,
fun-converges-to-pointwise,
sq_stable__req,
req_inversion,
req_weakening,
converges-to_functionality,
unique-limit,
nat_plus_wf,
icompact_wf,
i-approx_wf,
sq_stable__i-member,
imax_wf,
imax_nat_plus,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-less_than,
intformeq_wf,
int_formula_prop_eq_lemma,
i-approx-monotonic,
imax_ub,
sq_stable__icompact,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
istype-le,
subinterval_wf,
i-approx-containing2,
i-approx-closed,
i-approx-finite,
r-archimedean,
i-length_wf,
mul_nat_plus,
nat_properties,
itermAdd_wf,
int_term_value_add_lemma,
i-approx-is-subinterval,
rmin-rmax-subinterval,
istype-int_upper,
rleq_wf,
rabs_wf,
rsub_wf,
int_upper_properties,
subtype_rel_sets_simple,
rccint_wf,
rmin_wf,
rmax_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
integral_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
upper_subtype_nat,
sq_stable__le,
le_weakening2,
rmul_wf,
rsub_functionality,
I-norm_wf,
uimplies_transitivity,
rleq_functionality_wrt_implies,
rabs-integral,
rleq_weakening_equal,
rleq_functionality,
rabs_functionality,
integral-rsub,
I-norm-rleq,
mul_bounds_1b,
le_witness_for_triv,
rleq-int,
rleq_transitivity,
sq_stable__rleq,
i-member-diff-bound,
zero-rleq-rabs,
rleq-int-fractions2,
itermMultiply_wf,
int_term_value_mul_lemma,
rmul_functionality_wrt_rleq2,
rless_functionality,
rmul-int,
rmul_functionality,
rdiv_functionality,
rmul-is-positive,
rless_transitivity1,
rless_irreflexivity,
rless_transitivity2,
rleq_weakening_rless,
rmul_preserves_rleq,
rinv_wf2,
itermSubtract_wf,
req_transitivity,
rinv-of-rmul,
rmul-rinv,
rmul-rinv3,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
rleq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
setIsType,
universeIsType,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
functionIsType,
because_Cache,
setElimination,
rename,
applyEquality,
functionEquality,
setEquality,
lambdaEquality_alt,
dependent_functionElimination,
inhabitedIsType,
independent_functionElimination,
dependent_set_memberEquality_alt,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
productElimination,
independent_isectElimination,
dependent_pairFormation_alt,
natural_numberEquality,
unionElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
equalityIstype,
inrFormation_alt,
inlFormation_alt,
productIsType,
addEquality,
closedConclusion,
multiplyEquality,
isect_memberFormation_alt,
functionIsTypeImplies
Latex:
\mforall{}I:Interval. \mforall{}f:\mBbbN{} {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}. \mforall{}F:I {}\mrightarrow{}\mBbbR{}.
(lim n\mrightarrow{}\minfty{}.f[n;x] = \mlambda{}y.F[y] for x \mmember{} I
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}x,y:\{t:\mBbbR{}| t \mmember{} I\} . ((x = y) {}\mRightarrow{} (f[n;x] = f[n;y])))
{}\mRightarrow{} (\mforall{}a:\{a:\mBbbR{}| a \mmember{} I\} . lim n\mrightarrow{}\minfty{}.a\_\mint{}\msupminus{}x f[n;t] dt = \mlambda{}x.a\_\mint{}\msupminus{}x F[t] dt for x \mmember{} I))
Date html generated:
2019_10_30-AM-11_39_29
Last ObjectModification:
2019_04_09-PM-04_58_30
Theory : reals_2
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