Nuprl Lemma : Kummer-criterion
∀a,x:ℕ ⟶ ℝ.
((lim n→∞.a[n] * x[n] = r0
⇒ (∃c:{c:ℝ| r0 < c}
∃N:ℕ
((∀n:{N...}. ((r0 < a[n]) ∧ (r0 < x[n])))
∧ (∀n:{N...}. ((r0 < a[n]) ∧ (c ≤ ((a[n] * x[n]/x[n + 1]) - a[n + 1]))))))
⇒ Σn.x[n]↓)
∧ ((∃N:ℕ
((∀n:{N...}. ((r0 < a[n]) ∧ (r0 < x[n])))
∧ (∀n:{N...}. (((a[n] * x[n]/x[n + 1]) - a[n + 1]) ≤ r0))
∧ Σn.(r1/a[N + n])↑))
⇒ 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),
rleq: x ≤ y,
rless: x < y,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
int_upper: {i...},
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
nat: ℕ,
so_apply: x[s],
int_upper: {i...},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
so_lambda: λ2x.t[x],
series-converges: Σn.x[n]↓,
series-sum: Σn.x[n] = a,
converges: x[n]↓ as n→∞,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cauchy: cauchy(n.x[n]),
sq_stable: SqStable(P),
squash: ↓T,
rdiv: (x/y),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
converges-to: lim n→∞.x[n] = y,
rleq: x ≤ y,
rnonneg: rnonneg(x),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
subtract: n - m,
true: True,
int_seg: {i..j-},
lelt: i ≤ j < k,
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
sq_type: SQType(T),
real: ℝ
Lemmas referenced :
real_wf,
rless_wf,
int-to-real_wf,
istype-nat,
istype-int_upper,
int_upper_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
nat_plus_properties,
rleq_wf,
rsub_wf,
rdiv_wf,
itermAdd_wf,
int_term_value_add_lemma,
converges-to_wf,
rmul_wf,
series-diverges_wf,
converges-iff-cauchy-ext,
rsum_wf,
int_seg_subtype_nat,
istype-false,
int_seg_wf,
nat_plus_wf,
small-reciprocal-real,
rless-int,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
rmul_preserves_rless,
itermSubtract_wf,
itermMultiply_wf,
rinv_wf2,
sq_stable__rless,
rless_functionality,
req_transitivity,
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,
int_term_value_mul_lemma,
istype-less_than,
sq_stable__all,
nat_wf,
le_wf,
rabs_wf,
sq_stable__rleq,
le_witness_for_triv,
imax_wf,
sq_stable__less_than,
intformeq_wf,
int_formula_prop_eq_lemma,
imax_ub,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening,
radd_wf,
r-triangle-inequality2,
rless_functionality_wrt_implies,
rleq_functionality,
rabs-difference-symmetry,
req_weakening,
radd_functionality_wrt_rleq,
radd-rdiv,
rdiv_functionality,
radd-int,
req-int-fractions,
decidable__equal_int,
rless_transitivity2,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
rabs_functionality,
rsum-difference,
rsum_functionality_wrt_rleq,
subtype_rel_function,
subtype_rel_self,
subtract_wf,
int_seg_properties,
int_term_value_subtract_lemma,
subtract-add-cancel,
rmul_preserves_rleq,
rmul_preserves_rleq2,
rleq_weakening_rless,
rminus_wf,
itermMinus_wf,
radd_functionality,
rmul_functionality,
rmul-rinv,
real_term_value_add_lemma,
real_term_value_minus_lemma,
squash_wf,
true_wf,
iff_weakening_equal,
rsum_nonneg,
rabs-of-nonneg,
rleq_transitivity,
rsum_linearity2,
upper_subtype_nat,
le_transitivity,
add-subtract-cancel,
rsum-telescopes2,
rabs-bounds,
rinv-mul-as-rdiv,
rsum-zero-req,
rleq-int-fractions2,
radd-preserves-rleq,
ge_wf,
subtract-1-ge-0,
subtype_base_sq,
int_subtype_base,
trivial-int-eq1,
comparison-test-for-divergence,
series-diverges-rmul,
rmul-is-positive,
rmul-nonneg-case1,
series-diverges-tail
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
productIsType,
setIsType,
universeIsType,
introduction,
extract_by_obid,
hypothesis,
isectElimination,
natural_numberEquality,
hypothesisEquality,
functionIsType,
setElimination,
rename,
applyEquality,
dependent_set_memberEquality_alt,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
because_Cache,
addEquality,
inrFormation_alt,
closedConclusion,
inhabitedIsType,
imageMemberEquality,
baseClosed,
imageElimination,
multiplyEquality,
functionEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
dependent_set_memberFormation_alt,
applyLambdaEquality,
equalityIstype,
inlFormation_alt,
minusEquality,
promote_hyp,
instantiate,
universeEquality,
intWeakElimination,
cumulativity,
intEquality
Latex:
\mforall{}a,x:\mBbbN{} {}\mrightarrow{} \mBbbR{}.
((lim n\mrightarrow{}\minfty{}.a[n] * x[n] = r0
{}\mRightarrow{} (\mexists{}c:\{c:\mBbbR{}| r0 < c\}
\mexists{}N:\mBbbN{}
((\mforall{}n:\{N...\}. ((r0 < a[n]) \mwedge{} (r0 < x[n])))
\mwedge{} (\mforall{}n:\{N...\}. ((r0 < a[n]) \mwedge{} (c \mleq{} ((a[n] * x[n]/x[n + 1]) - a[n + 1]))))))
{}\mRightarrow{} \mSigma{}n.x[n]\mdownarrow{})
\mwedge{} ((\mexists{}N:\mBbbN{}
((\mforall{}n:\{N...\}. ((r0 < a[n]) \mwedge{} (r0 < x[n])))
\mwedge{} (\mforall{}n:\{N...\}. (((a[n] * x[n]/x[n + 1]) - a[n + 1]) \mleq{} r0))
\mwedge{} \mSigma{}n.(r1/a[N + n])\muparrow{}))
{}\mRightarrow{} \mSigma{}n.x[n]\muparrow{}))
Date html generated:
2019_10_29-AM-10_27_35
Last ObjectModification:
2018_12_13-PM-02_06_23
Theory : reals
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