Nuprl Lemma : not-diverges-converges
∀[x:ℕ ⟶ ℝ]. (¬(x[n]↓ as n→∞ ∧ n.x[n]↑))
Proof
Definitions occuring in Statement :
diverges: n.x[n]↑,
converges: x[n]↓ as n→∞,
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
not: ¬A,
and: P ∧ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
cauchy: cauchy(n.x[n]),
diverges: n.x[n]↑,
exists: ∃x:A. B[x],
prop: ℙ,
sq_exists: ∃x:{A| B[x]},
guard: {T},
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
or: P ∨ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
rless_irreflexivity,
rless_transitivity1,
rsub_wf,
rabs_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
rless-int,
rdiv_wf,
int-to-real_wf,
rless_wf,
small-reciprocal-real,
real_wf,
diverges_wf,
converges_wf,
and_wf,
nat_wf,
converges-iff-cauchy
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
sqequalHypSubstitution,
productElimination,
lemma_by_obid,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesisEquality,
hypothesis,
independent_functionElimination,
because_Cache,
voidElimination,
isectElimination,
functionEquality,
dependent_set_memberEquality,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
inrFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}[x:\mBbbN{} {}\mrightarrow{} \mBbbR{}]. (\mneg{}(x[n]\mdownarrow{} as n\mrightarrow{}\minfty{} \mwedge{} n.x[n]\muparrow{}))
Date html generated:
2016_05_18-AM-07_39_29
Last ObjectModification:
2016_01_17-AM-02_04_15
Theory : reals
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