Nuprl Lemma : Raabe-test
∀x:ℕ ⟶ ℝ. ∀L:ℝ.
((∀n:ℕ. (r0 < x[n]))
⇒ lim n→∞.r(n) * ((x[n]/x[n + 1]) - r1) = L
⇒ (((r1 < L) ⇒ Σn.x[n]↓) ∧ ((L < r1) ⇒ Σn.x[n]↑)))
Proof
Definitions occuring in Statement :
series-diverges: Σn.x[n]↑,
series-converges: Σn.x[n]↓,
converges-to: lim n→∞.x[n] = y,
rdiv: (x/y),
rless: x < y,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: 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: ℕ,
so_apply: x[s],
and: P ∧ Q,
implies: P ⇒ Q,
cand: A c∧ B,
uimplies: b supposing a,
prop: ℙ,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rless: x < y,
sq_exists: ∃x:A [B[x]],
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
ge: i ≥ j ,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
itermSubtract: left (-) right,
int_term_ind: int_term_ind,
real_term_value: real_term_value(f;t),
itermConstant: "const",
rev_uimplies: rev_uimplies(P;Q),
true: True,
rsub: x - y,
squash: ↓T,
subtype_rel: A ⊆r B,
real: ℝ,
sq_stable: SqStable(P),
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermMultiply: left "*" right,
rtermVar: rtermVar(var),
pi1: fst(t),
pi2: snd(t),
rdiv: (x/y),
converges-to: lim n→∞.x[n] = y,
le: A ≤ B,
subtract: n - m,
less_than': less_than'(a;b),
rleq: x ≤ y,
rnonneg: rnonneg(x),
int_upper: {i...},
rge: x ≥ y
Lemmas referenced :
Kummer-criterion,
int-to-real_wf,
istype-nat,
small-reciprocal-real,
rless-implies-rless,
rsub_wf,
rless_wf,
rdiv_wf,
rless-int,
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,
converges-to_wf,
rmul_wf,
nat_properties,
decidable__le,
intformle_wf,
itermAdd_wf,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
istype-le,
real_wf,
itermSubtract_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rmul-limit,
rsub-limit,
nat_wf,
satisfiable-full-omega-tt,
constant-limit,
req_weakening,
rinv-converges-to-0,
real_term_polynomial,
converges-to_functionality,
rmul_preserves_req,
req_wf,
radd_wf,
rminus_wf,
req-int,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
uiff_transitivity,
req_functionality,
rmul-rdiv-cancel2,
req_transitivity,
rmul-distrib,
radd_functionality,
rmul_over_rminus,
rmul-one-both,
rminus_functionality,
uiff_transitivity3,
squash_wf,
true_wf,
rminus-int,
radd-int,
sq_stable__less_than,
rmul-identity1,
rmul-is-positive,
rmul-rsub-distrib,
rsub_functionality,
rmul_functionality,
rmul-assoc,
rmul_comm,
req_inversion,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermDivide_wf,
rtermVar_wf,
radd-preserves-req,
rinv_wf2,
itermMinus_wf,
itermMultiply_wf,
rminus-rdiv,
rmul-int-rdiv,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_mul_lemma,
rmul-int,
rmul-rinv,
Raabe-lemma,
istype-false,
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,
istype-less_than,
istype-int_upper,
le_witness_for_triv,
upper_subtype_nat,
not-le-2,
sq_stable__le,
add-swap,
int_upper_properties,
rleq_wf,
rabs-difference-bound-rleq,
rabs_wf,
sq_stable__rleq,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening,
radd-preserves-rleq,
rleq_functionality,
rinv-as-rdiv,
rinv-mul-as-rdiv,
series-diverges_wf,
rneq-int,
radd-preserves-rless,
rless_functionality,
rleq_weakening_rless,
series-diverges-tail-iff,
harmonic-series-diverges,
series-diverges_functionality,
req-int-fractions,
int_term_value_mul_lemma,
set_subtype_base,
le_wf,
int_subtype_base
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
sqequalRule,
lambdaEquality_alt,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
lambdaFormation_alt,
productElimination,
independent_functionElimination,
natural_numberEquality,
independent_isectElimination,
dependent_set_memberEquality_alt,
because_Cache,
universeIsType,
inhabitedIsType,
closedConclusion,
inrFormation_alt,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
applyEquality,
addEquality,
functionIsType,
lambdaEquality,
inrFormation,
dependent_pairFormation,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
lambdaFormation,
minusEquality,
imageElimination,
imageMemberEquality,
baseClosed,
inlFormation_alt,
productIsType,
equalityIsType1,
multiplyEquality,
functionIsTypeImplies,
equalityIsType4,
baseApply
Latex:
\mforall{}x:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}L:\mBbbR{}.
((\mforall{}n:\mBbbN{}. (r0 < x[n]))
{}\mRightarrow{} lim n\mrightarrow{}\minfty{}.r(n) * ((x[n]/x[n + 1]) - r1) = L
{}\mRightarrow{} (((r1 < L) {}\mRightarrow{} \mSigma{}n.x[n]\mdownarrow{}) \mwedge{} ((L < r1) {}\mRightarrow{} \mSigma{}n.x[n]\muparrow{})))
Date html generated:
2019_10_29-AM-10_28_55
Last ObjectModification:
2019_04_02-AM-10_00_22
Theory : reals
Home
Index