Nuprl Lemma : rinv-exp-converges
∀M:ℕ+. ∀N:{2...}. lim n→∞.(r1/r(M * N^n)) = r0
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
rdiv: (x/y),
int-to-real: r(n),
exp: i^n,
int_upper: {i...},
nat_plus: ℕ+,
all: ∀x:A. B[x],
multiply: n * m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
converges-to: lim n→∞.x[n] = y,
member: t ∈ T,
subtype_rel: A ⊆r B,
sq_exists: ∃x:{A| B[x]},
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
int_upper: {i...},
prop: ℙ,
implies: P ⇒ Q,
nat: ℕ,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
decidable: Dec(P),
not: ¬A,
false: False,
uiff: uiff(P;Q),
top: Top,
less_than': less_than'(a;b),
true: True,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
so_apply: x[s],
rless: x < y,
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
sq_stable: SqStable(P),
squash: ↓T,
real: ℝ,
less_than: a < b
Lemmas referenced :
exp-ratio-property2,
nat_plus_subtype_nat,
exp-ratio_wf2,
nat_wf,
less_than_wf,
exp_wf2,
le_wf,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
mul_bounds_1b,
exp_wf_nat_plus,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
rless_wf,
nat_properties,
nat_plus_properties,
int_upper_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
nat_plus_wf,
int_upper_wf,
rmul_wf,
rinv_wf2,
uiff_transitivity,
rleq_functionality,
rabs_functionality,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rinv-as-rdiv,
le_weakening,
le_weakening2,
le_functionality,
sq_stable__le,
equal_wf,
int_term_value_subtract_lemma,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
real_wf,
sq_stable__less_than,
subtract_wf,
trivial-int-eq1,
iff_weakening_equal,
exp_add,
true_wf,
squash_wf,
exp_preserves_le,
int_term_value_mul_lemma,
multiply-is-int-iff,
int_upper_subtype_nat,
exp_wf4,
mul_preserves_le,
rleq-int-fractions2,
int_subtype_base,
less_than_transitivity1,
rleq-int-fractions,
rabs-of-nonneg,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
dependent_set_memberFormation,
lambdaEquality,
setElimination,
rename,
setEquality,
isectElimination,
multiplyEquality,
because_Cache,
functionEquality,
natural_numberEquality,
independent_isectElimination,
inrFormation,
productElimination,
independent_functionElimination,
dependent_set_memberEquality,
unionElimination,
independent_pairFormation,
voidElimination,
isect_memberEquality,
voidEquality,
intEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
equalitySymmetry,
equalityTransitivity,
imageElimination,
baseClosed,
imageMemberEquality,
applyLambdaEquality,
addEquality,
universeEquality,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality
Latex:
\mforall{}M:\mBbbN{}\msupplus{}. \mforall{}N:\{2...\}. lim n\mrightarrow{}\minfty{}.(r1/r(M * N\^{}n)) = r0
Date html generated:
2017_10_03-AM-08_54_37
Last ObjectModification:
2017_07_28-AM-07_36_28
Theory : reals
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