Nuprl Lemma : series-diverges-trivially
∀z:ℝ. ((r0 < z) ⇒ (∀x:ℕ ⟶ ℝ. ((∀k:ℕ. ∃n:ℕ. ((k ≤ n) ∧ (z ≤ |x[n]|))) ⇒ Σn.x[n]↑)))
Proof
Definitions occuring in Statement :
series-diverges: Σn.x[n]↑,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
so_apply: x[s],
le: A ≤ B,
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 :
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
subtype_rel: A ⊆r B,
int_upper: {i...},
less_than': less_than'(a;b),
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
bfalse: ff,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
nat_plus: ℕ+,
squash: ↓T,
sq_stable: SqStable(P),
sq_exists: ∃x:A [B[x]],
rless: x < y,
uall: ∀[x:A]. B[x],
nat: ℕ,
cand: A c∧ B,
and: P ∧ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
diverges: n.x[n]↑,
series-diverges: Σn.x[n]↑,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rsum_unroll,
rsub_functionality,
rabs_functionality,
req_weakening,
rleq_functionality,
req-iff-rsub-is-0,
int_formula_prop_less_lemma,
intformless_wf,
int_seg_subtype_nat,
radd_wf,
zero-add,
nequal-le-implies,
false_wf,
upper_subtype_nat,
neg_assert_of_eq_int,
int_formula_prop_eq_lemma,
intformeq_wf,
assert_of_eq_int,
eq_int_wf,
less_than_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
real_wf,
all_wf,
int-to-real_wf,
rless_wf,
nat_wf,
exists_wf,
int_seg_wf,
rsum_wf,
rsub_wf,
rabs_wf,
rleq_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_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,
full-omega-unsat,
decidable__le,
nat_plus_properties,
sq_stable__less_than,
nat_properties
Rules used in proof :
hypothesis_subsumption,
functionExtensionality,
cumulativity,
instantiate,
promote_hyp,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
functionEquality,
applyEquality,
productEquality,
productElimination,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
approximateComputation,
independent_isectElimination,
unionElimination,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
independent_functionElimination,
because_Cache,
isectElimination,
extract_by_obid,
introduction,
natural_numberEquality,
rename,
setElimination,
addEquality,
dependent_set_memberEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
independent_pairFormation,
hypothesis,
cut,
hypothesisEquality,
dependent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}z:\mBbbR{}. ((r0 < z) {}\mRightarrow{} (\mforall{}x:\mBbbN{} {}\mrightarrow{} \mBbbR{}. ((\mforall{}k:\mBbbN{}. \mexists{}n:\mBbbN{}. ((k \mleq{} n) \mwedge{} (z \mleq{} |x[n]|))) {}\mRightarrow{} \mSigma{}n.x[n]\muparrow{})))
Date html generated:
2018_05_22-PM-02_02_41
Last ObjectModification:
2018_05_21-AM-00_16_18
Theory : reals
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