Nuprl Lemma : ratio-test-corollary
∀x:ℕ ⟶ ℝ
((∀n:ℕ. x[n] ≠ r0) ⇒ (∀L:ℝ. (lim n→∞.|(x[n + 1]/x[n])| = L ⇒ (((L < r1) ⇒ Σn.x[n]↓) ∧ ((r1 < L) ⇒ 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),
rneq: x ≠ y,
rless: x < y,
rabs: |x|,
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],
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
rneq: x ≠ y,
guard: {T},
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
real: ℝ,
sq_stable: SqStable(P),
squash: ↓T,
rdiv: (x/y),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
converges-to: lim n→∞.x[n] = y,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
int_upper: {i...},
itermConstant: "const",
true: True,
less_than': less_than'(a;b),
subtract: n - m
Lemmas referenced :
ratio-test-ext,
rless_wf,
int-to-real_wf,
converges-to_wf,
rabs_wf,
rdiv_wf,
nat_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
real_wf,
all_wf,
rneq_wf,
small-reciprocal-real,
rless-implies-rless,
rsub_wf,
itermSubtract_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
zero-rleq-rabs,
sq_stable__less_than,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformless_wf,
int_formula_prop_less_lemma,
constant-rleq-limit,
radd-preserves-rless,
radd_wf,
rmul_wf,
rinv_wf2,
itermMultiply_wf,
rless_functionality,
req_transitivity,
radd_functionality,
req_weakening,
rinv-as-rdiv,
real_term_value_add_lemma,
real_term_value_mul_lemma,
trivial-rleq-radd,
rleq-int-fractions2,
int_term_value_mul_lemma,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
sq_stable__all,
rleq_wf,
sq_stable__rleq,
less_than'_wf,
nat_plus_wf,
squash_wf,
int_upper_wf,
upper_subtype_nat,
sq_stable__le,
int_upper_properties,
rpositive-rless,
rabs-positive,
rabs-difference-bound-rleq,
rleq_functionality,
rabs_functionality,
rsub_functionality,
rabs-rdiv,
rmul_preserves_rleq2,
rleq_weakening_rless,
equal_wf,
rmul_comm,
rleq-implies-rleq,
rmul_functionality,
radd_comm,
rmul-rinv3,
real_term_polynomial,
less_than_wf,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
less-iff-le,
not-lt-2,
false_wf,
int_upper_subtype_nat,
rless-int-fractions,
rminus_wf,
itermMinus_wf,
real_term_value_minus_lemma,
rmul_preserves_rless,
rless_transitivity2,
radd-preserves-rleq
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
natural_numberEquality,
independent_pairFormation,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
functionEquality,
productElimination,
inrFormation,
computeAll,
imageMemberEquality,
baseClosed,
imageElimination,
multiplyEquality,
equalityTransitivity,
equalitySymmetry,
independent_pairEquality,
minusEquality,
axiomEquality,
productEquality,
isect_memberFormation
Latex:
\mforall{}x:\mBbbN{} {}\mrightarrow{} \mBbbR{}
((\mforall{}n:\mBbbN{}. x[n] \mneq{} r0)
{}\mRightarrow{} (\mforall{}L:\mBbbR{}. (lim n\mrightarrow{}\minfty{}.|(x[n + 1]/x[n])| = L {}\mRightarrow{} (((L < r1) {}\mRightarrow{} \mSigma{}n.x[n]\mdownarrow{}) \mwedge{} ((r1 < L) {}\mRightarrow{} \mSigma{}n.x[n]\muparrow{})))))
Date html generated:
2019_10_29-AM-10_27_03
Last ObjectModification:
2018_08_23-AM-11_25_24
Theory : reals
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