Nuprl Lemma : req-iff-rational-approx
∀[x:ℝ]. ∀[a:ℕ+ ⟶ ℤ]. (regular-seq(a) ∧ (a = x) ⇐⇒ ∀n:ℕ+. (|x - (r(a n)/r(2 * n))| ≤ (r1/r(n))))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
req: x = y,
int-to-real: r(n),
real: ℝ,
regular-int-seq: k-regular-seq(f),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: 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],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
real: ℝ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
so_apply: x[s],
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rational-approx: (x within 1/n),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
cand: A c∧ B
Lemmas referenced :
regular-int-seq_wf,
req_wf,
nat_plus_wf,
all_wf,
rleq_wf,
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,
real_wf,
less_than'_wf,
rational-approx-property,
rleq_functionality,
rabs_functionality,
rsub_functionality,
req_inversion,
req_weakening,
int-rdiv_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-base,
int_subtype_base,
nequal_wf,
int-rdiv-req,
rational-approx-implies-req,
less_than_wf,
accelerate-req,
accelerate_wf,
req_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
productEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
functionExtensionality,
applyEquality,
hypothesisEquality,
because_Cache,
hypothesis,
dependent_set_memberEquality,
sqequalRule,
lambdaEquality,
multiplyEquality,
setElimination,
rename,
independent_isectElimination,
inrFormation,
dependent_functionElimination,
productElimination,
independent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
functionEquality,
independent_pairEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
baseApply,
closedConclusion,
baseClosed,
imageMemberEquality
Latex:
\mforall{}[x:\mBbbR{}]. \mforall{}[a:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}]. (regular-seq(a) \mwedge{} (a = x) \mLeftarrow{}{}\mRightarrow{} \mforall{}n:\mBbbN{}\msupplus{}. (|x - (r(a n)/r(2 * n))| \mleq{} (r1/r(n))))
Date html generated:
2018_05_22-PM-01_58_24
Last ObjectModification:
2017_10_26-PM-03_33_17
Theory : reals
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