Nuprl Lemma : series-converges-tail
∀x:ℕ ⟶ ℝ. (Σn.x[n]↓ ⇒ (∀y:ℕ ⟶ ℝ. ((∃N:ℕ. ∀n:{N...}. (y[n] = x[n])) ⇒ Σn.y[n]↓)))
Proof
Definitions occuring in Statement :
series-converges: Σn.x[n]↓,
req: x = y,
real: ℝ,
int_upper: {i...},
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
series-converges: Σn.x[n]↓,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
series-sum: Σn.x[n] = a,
converges-to: lim n→∞.x[n] = y,
sq_exists: ∃x:{A| B[x]},
guard: {T},
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
int_seg: {i..j-},
lelt: i ≤ j < k,
uiff: uiff(P;Q),
int_upper: {i...},
pointwise-req: x[k] = y[k] for k ∈ [n,m],
rsub: x - y,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
real_term_value: real_term_value(f;t),
int_term_ind: int_term_ind,
itermSubtract: left (-) right,
itermVar: vvar,
itermAdd: left (+) right,
itermMultiply: left (*) right,
itermMinus: "-"num,
rge: x ≥ y
Lemmas referenced :
radd_wf,
rsum_wf,
rsub_wf,
nat_wf,
int_seg_subtype_nat,
false_wf,
int_seg_wf,
imax_wf,
imax_nat,
nat_properties,
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,
le_wf,
all_wf,
rleq_wf,
rabs_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
rless_wf,
nat_plus_wf,
series-sum_wf,
exists_wf,
int_upper_wf,
req_wf,
int_upper_subtype_nat,
real_wf,
series-converges_wf,
imax_ub,
le_functionality,
le_weakening,
rsum-split,
subtype_rel_dep_function,
subtype_rel_self,
int_seg_properties,
itermAdd_wf,
int_term_value_add_lemma,
req_functionality,
req_weakening,
rsub_functionality,
rsum_functionality,
radd_functionality,
uiff_transitivity,
rminus_wf,
radd-preserves-req,
req_inversion,
radd-assoc,
radd_comm,
req_transitivity,
radd-ac,
rminus_functionality,
rsum_functionality2,
radd-rminus-assoc,
rmul_wf,
rminus-as-rmul,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermMinus_wf,
req-iff-rsub-is-0,
rsum_linearity2,
rsum_linearity1,
rabs_functionality,
rleq_functionality,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rmul_functionality,
rminus-radd
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
addEquality,
independent_isectElimination,
independent_pairFormation,
dependent_functionElimination,
dependent_set_memberFormation,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
functionEquality,
inrFormation,
inlFormation,
minusEquality
Latex:
\mforall{}x:\mBbbN{} {}\mrightarrow{} \mBbbR{}. (\mSigma{}n.x[n]\mdownarrow{} {}\mRightarrow{} (\mforall{}y:\mBbbN{} {}\mrightarrow{} \mBbbR{}. ((\mexists{}N:\mBbbN{}. \mforall{}n:\{N...\}. (y[n] = x[n])) {}\mRightarrow{} \mSigma{}n.y[n]\mdownarrow{})))
Date html generated:
2017_10_03-AM-09_18_39
Last ObjectModification:
2017_07_28-AM-07_43_35
Theory : reals
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