Nuprl Lemma : series-diverges-tail-iff
∀x:ℕ ⟶ ℝ. ∀N:ℕ. (Σn.x[N + n]↑ ⇐⇒ Σn.x[n]↑)
Proof
Definitions occuring in Statement :
series-diverges: Σn.x[n]↑,
real: ℝ,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
add: n + m
Definitions unfolded in proof :
pointwise-req: x[k] = y[k] for k ∈ [n,m],
uiff: uiff(P;Q),
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
le: A ≤ B,
nat_plus: ℕ+,
squash: ↓T,
sq_stable: SqStable(P),
real: ℝ,
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
sq_exists: ∃x:{A| B[x]},
rless: x < y,
guard: {T},
int_seg: {i..j-},
cand: A c∧ B,
diverges: n.x[n]↑,
series-diverges: Σn.x[n]↑,
rev_implies: P ⇐ Q,
top: Top,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
rsum-difference,
req_weakening,
add-commutes,
rsum_functionality,
rabs_functionality,
rabs-difference-symmetry,
req_functionality,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rleq_weakening,
rsum-shift,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
int_seg_wf,
nat_plus_properties,
real_wf,
sq_stable__less_than,
int_seg_properties,
rsum_wf,
rsub_wf,
rabs_wf,
rleq_wf,
exists_wf,
all_wf,
int-to-real_wf,
rless_wf,
le_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
nat_wf,
series-diverges_wf,
series-diverges-tail
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
functionEquality,
imageElimination,
baseClosed,
imageMemberEquality,
because_Cache,
productEquality,
promote_hyp,
productElimination,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_isectElimination,
unionElimination,
natural_numberEquality,
rename,
setElimination,
addEquality,
dependent_set_memberEquality,
functionExtensionality,
applyEquality,
lambdaEquality,
sqequalRule,
isectElimination,
independent_functionElimination,
independent_pairFormation,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}x:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}N:\mBbbN{}. (\mSigma{}n.x[N + n]\muparrow{} \mLeftarrow{}{}\mRightarrow{} \mSigma{}n.x[n]\muparrow{})
Date html generated:
2016_11_08-AM-09_01_25
Last ObjectModification:
2016_11_07-AM-11_54_08
Theory : reals
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