Nuprl Lemma : fun-series-converges-tail
∀M:ℕ. ∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ. (Σn.f[n + M;x]↓ for x ∈ I ⇒ Σn.f[n;x]↓ for x ∈ I)
Proof
Definitions occuring in Statement :
fun-series-converges: Σn.f[n; x]↓ for x ∈ I,
rfun: I ⟶ℝ,
interval: Interval,
nat: ℕ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
add: n + m
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
fun-series-converges: Σn.f[n; x]↓ for x ∈ I,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
int_seg: {i..j-},
guard: {T},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
iff: P ⇐⇒ Q,
le: A ≤ B,
less_than': less_than'(a;b),
rev_implies: P ⇐ Q,
fun-cauchy: λn.f[n; x] is cauchy for x ∈ I,
nat_plus: ℕ+,
int_upper: {i...},
rneq: x ≠ y,
label: ...$L... t,
sq_stable: SqStable(P),
squash: ↓T,
subinterval: I ⊆ J ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
fun-converges-iff-cauchy,
rsum_wf,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
int_seg_wf,
real_wf,
i-member_wf,
int_seg_subtype_nat,
istype-false,
nat_plus_properties,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
istype-int_upper,
i-approx_wf,
rleq_wf,
rabs_wf,
rsub_wf,
i-member-approx,
rdiv_wf,
int-to-real_wf,
rless-int,
int_upper_properties,
rless_wf,
nat_plus_wf,
icompact_wf,
fun-series-converges_wf,
rfun_wf,
interval_wf,
istype-nat,
subtract_wf,
sq_stable__icompact,
itermSubtract_wf,
int_term_value_subtract_lemma,
i-approx-is-subinterval,
subtype_rel_sets_simple,
rleq_functionality,
rabs_functionality,
rsum-difference,
req_weakening,
general_arith_equation1,
rsum-shift,
rabs-difference-symmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
isectElimination,
closedConclusion,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
applyEquality,
dependent_set_memberEquality_alt,
addEquality,
productElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
setIsType,
inhabitedIsType,
functionIsType,
inrFormation_alt,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}M:\mBbbN{}. \mforall{}I:Interval. \mforall{}f:\mBbbN{} {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}. (\mSigma{}n.f[n + M;x]\mdownarrow{} for x \mmember{} I {}\mRightarrow{} \mSigma{}n.f[n;x]\mdownarrow{} for x \mmember{} I)
Date html generated:
2019_10_30-AM-08_59_17
Last ObjectModification:
2018_11_08-PM-02_13_18
Theory : reals
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