Nuprl Lemma : comparison-test-for-divergence
∀x,y:ℕ ⟶ ℝ. (Σn.y[n]↑ ⇒ (∃N:ℕ. ∀n:{N...}. ((r0 ≤ y[n]) ∧ (y[n] ≤ x[n]))) ⇒ Σn.x[n]↑)
Proof
Definitions occuring in Statement :
series-diverges: Σn.x[n]↑,
rleq: x ≤ y,
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,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
series-diverges: Σn.x[n]↑,
diverges: n.x[n]↑,
exists: ∃x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
prop: ℙ,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
guard: {T},
rless: x < y,
sq_exists: ∃x:{A| B[x]},
real: ℝ,
sq_stable: SqStable(P),
squash: ↓T,
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
int_seg: {i..j-},
lelt: i ≤ j < k,
uiff: uiff(P;Q),
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
int_upper: {i...}
Lemmas referenced :
nat_wf,
rless_wf,
int-to-real_wf,
all_wf,
exists_wf,
le_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rsum_wf,
int_seg_subtype_nat,
false_wf,
int_seg_wf,
int_upper_wf,
int_upper_subtype_nat,
series-diverges_wf,
real_wf,
imax_wf,
imax_nat,
nat_properties,
sq_stable__less_than,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
equal_wf,
imax_ub,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
int_seg_properties,
itermAdd_wf,
int_term_value_add_lemma,
rleq_functionality,
rabs_functionality,
rsum-difference,
rabs-of-nonneg,
rsum_nonneg,
rleq_transitivity,
rsum_functionality_wrt_rleq,
rabs-difference-symmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
independent_pairFormation,
promote_hyp,
hypothesis,
cut,
introduction,
extract_by_obid,
sqequalRule,
productEquality,
isectElimination,
natural_numberEquality,
lambdaEquality,
because_Cache,
setElimination,
rename,
applyEquality,
functionExtensionality,
addEquality,
independent_isectElimination,
functionEquality,
dependent_functionElimination,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
inrFormation,
inlFormation
Latex:
\mforall{}x,y:\mBbbN{} {}\mrightarrow{} \mBbbR{}. (\mSigma{}n.y[n]\muparrow{} {}\mRightarrow{} (\mexists{}N:\mBbbN{}. \mforall{}n:\{N...\}. ((r0 \mleq{} y[n]) \mwedge{} (y[n] \mleq{} x[n]))) {}\mRightarrow{} \mSigma{}n.x[n]\muparrow{})
Date html generated:
2017_10_03-AM-09_19_59
Last ObjectModification:
2017_07_28-AM-07_44_41
Theory : reals
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