Nuprl Lemma : rational-approx-implies-req
∀[k:ℕ+]. ∀[x:ℝ]. ∀[a:ℕ+ ⟶ ℤ].
((∀n:ℕ+. (|x - (r(a n)/r(2 * n))| ≤ (r(k)/r(n)))) ⇒ {k-regular-seq(a) ∧ (accelerate(k;a) = x)})
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
req: x = y,
int-to-real: r(n),
accelerate: accelerate(k;f),
real: ℝ,
regular-int-seq: k-regular-seq(f),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
multiply: n * m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
guard: {T},
all: ∀x:A. B[x],
rev_uimplies: rev_uimplies(P;Q),
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
rge: x ≥ y,
uiff: uiff(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
regular-int-seq: k-regular-seq(f),
le: A ≤ B,
subtype_rel: A ⊆r B,
nat: ℕ,
req_int_terms: t1 ≡ t2,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
rless: x < y,
sq_exists: ∃x:A [B[x]],
real: ℝ,
sq_stable: SqStable(P),
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
int_nzero: ℤ-o,
rdiv: (x/y),
bdd-diff: bdd-diff(f;g),
ge: i ≥ j ,
rational-approx: (x within 1/n)
Lemmas referenced :
rleq_functionality_wrt_implies,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMultiply_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_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
rleq_weakening_equal,
rleq_weakening,
radd_wf,
r-triangle-inequality2,
radd_functionality_wrt_rleq,
rleq_functionality,
rabs-difference-symmetry,
req_weakening,
nat_plus_wf,
all_wf,
rleq_wf,
less_than'_wf,
absval_wf,
subtract_wf,
nat_wf,
req_witness,
accelerate_wf,
regular-int-seq_wf,
real_wf,
itermSubtract_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rleq-int,
rmul_wf,
rdiv-is-positive,
rmul_preserves_rleq,
squash_wf,
true_wf,
rneq_wf,
rabs-int,
iff_weakening_equal,
absval_pos,
le_wf,
mul_bounds_1a,
false_wf,
nat_plus_subtype_nat,
mul_bounds_1b,
less_than_wf,
mul_nat_plus,
req_transitivity,
rabs_functionality,
req_inversion,
rsub-int,
rsub_functionality,
rmul-int,
rmul_functionality,
radd-int,
rless_functionality,
rabs-of-nonneg,
uiff_transitivity,
rabs-rdiv,
rabs-rmul,
rleq-int-fractions2,
sq_stable__less_than,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
req_wf,
radd_functionality,
rinv_wf2,
rneq_functionality,
rneq-int,
int_entire_a,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
mul_nzero,
subtype_rel_sets,
nequal_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-T-base,
req_functionality,
rmul-distrib,
rinv_functionality2,
rinv-of-rmul,
rmul-rinv3,
real_term_value_mul_lemma,
rmul-rinv,
rmul-ac,
rmul-rsub-distrib,
rinv-mul-as-rdiv,
itermAdd_wf,
uimplies_transitivity,
real_term_value_add_lemma,
req-iff-rabs-rleq-bound,
accelerate-bdd-diff,
nat_properties,
uiff_transitivity2,
rsub-rdiv,
rdiv_functionality,
set_subtype_base,
absval-non-neg,
equal_wf,
mul_preserves_le,
rleq-int-fractions,
multiply-is-int-iff,
int_term_value_add_lemma,
int-rdiv_wf,
rational-approx_wf,
rational-approx-property,
int-rdiv-req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
functionExtensionality,
because_Cache,
hypothesis,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
sqequalRule,
inrFormation,
dependent_functionElimination,
productElimination,
independent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
independent_pairEquality,
addEquality,
axiomEquality,
dependent_set_memberEquality,
functionEquality,
inlFormation,
imageMemberEquality,
baseClosed,
productEquality,
imageElimination,
universeEquality,
addLevel,
instantiate,
cumulativity,
setEquality,
applyLambdaEquality,
baseApply,
closedConclusion
Latex:
\mforall{}[k:\mBbbN{}\msupplus{}]. \mforall{}[x:\mBbbR{}]. \mforall{}[a:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}].
((\mforall{}n:\mBbbN{}\msupplus{}. (|x - (r(a n)/r(2 * n))| \mleq{} (r(k)/r(n)))) {}\mRightarrow{} \{k-regular-seq(a) \mwedge{} (accelerate(k;a) = x)\})
Date html generated:
2018_05_22-PM-01_58_13
Last ObjectModification:
2017_10_26-PM-03_23_18
Theory : reals
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