Nuprl Lemma : fun-ratio-test-everywhere
∀f:ℕ ⟶ ℝ ⟶ ℝ
((∀n:ℕ. ∀x,y:ℝ. ((x = y) ⇒ (f[n;x] = f[n;y])))
⇒ (∀m:ℕ+. ∃c:ℝ. ((r0 ≤ c) ∧ (c < r1) ∧ (∃N:ℕ. ∀n:{N...}. ∀x:{x:ℝ| |x| ≤ r(m)} . (|f[n + 1;x]| ≤ (c * |f[n;x]|)))))
⇒ Σn.f[n;x]↓ absolutely for x ∈ (-∞, ∞))
Proof
Definitions occuring in Statement :
fun-series-converges-absolutely: Σn.f[n; x]↓ absolutely for x ∈ I,
riiint: (-∞, ∞),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
int_upper: {i...},
nat_plus: ℕ+,
nat: ℕ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
rfun: I ⟶ℝ,
top: Top,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
implies: P ⇒ Q,
so_apply: x[s1;s2],
guard: {T},
and: P ∧ Q,
nat: ℕ,
nat_plus: ℕ+,
int_upper: {i...},
rless: x < y,
sq_exists: ∃x:{A| B[x]},
real: ℝ,
sq_stable: SqStable(P),
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
cand: A c∧ B,
le: A ≤ B,
i-approx: i-approx(I;n),
riiint: (-∞, ∞),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True
Lemmas referenced :
fun-ratio-test,
riiint_wf,
member_riiint_lemma,
subtype_rel_dep_function,
real_wf,
true_wf,
subtype_rel_self,
set_wf,
nat_wf,
iproper-riiint,
i-member_wf,
req_wf,
nat_plus_wf,
icompact_wf,
i-approx_wf,
all_wf,
exists_wf,
rleq_wf,
int-to-real_wf,
rless_wf,
int_upper_wf,
rabs_wf,
int_upper_properties,
nat_properties,
sq_stable__less_than,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
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,
le_wf,
rmul_wf,
int_upper_subtype_nat,
less_than_wf,
sq_stable__icompact,
member_rccint_lemma,
rabs-rleq-iff,
squash_wf,
rminus-int,
iff_weakening_equal
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
hypothesis,
lambdaFormation,
functionExtensionality,
applyEquality,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
isectElimination,
lambdaEquality,
setEquality,
independent_isectElimination,
setElimination,
rename,
because_Cache,
independent_functionElimination,
functionEquality,
productEquality,
natural_numberEquality,
dependent_set_memberEquality,
addEquality,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
productElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality
Latex:
\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbR{} {}\mrightarrow{} \mBbbR{}
((\mforall{}n:\mBbbN{}. \mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (f[n;x] = f[n;y])))
{}\mRightarrow{} (\mforall{}m:\mBbbN{}\msupplus{}
\mexists{}c:\mBbbR{}
((r0 \mleq{} c)
\mwedge{} (c < r1)
\mwedge{} (\mexists{}N:\mBbbN{}. \mforall{}n:\{N...\}. \mforall{}x:\{x:\mBbbR{}| |x| \mleq{} r(m)\} . (|f[n + 1;x]| \mleq{} (c * |f[n;x]|)))))
{}\mRightarrow{} \mSigma{}n.f[n;x]\mdownarrow{} absolutely for x \mmember{} (-\minfty{}, \minfty{}))
Date html generated:
2016_10_26-AM-11_14_31
Last ObjectModification:
2016_08_28-PM-02_02_49
Theory : reals
Home
Index