Nuprl Lemma : rational-inner-approx-int
∀x:ℝ. ∀n:ℕ+. ∃z:ℤ. ((|(r(z)/r(4 * n))| ≤ |x|) ∧ (|x - (r(z)/r(4 * n))| ≤ (r(2)/r(n))))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
multiply: n * m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
prop: ℙ,
real: ℝ,
rational-inner-approx: rational-inner-approx(x;n),
has-value: (a)↓,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B
Latex:
\mforall{}x:\mBbbR{}. \mforall{}n:\mBbbN{}\msupplus{}. \mexists{}z:\mBbbZ{}. ((|(r(z)/r(4 * n))| \mleq{} |x|) \mwedge{} (|x - (r(z)/r(4 * n))| \mleq{} (r(2)/r(n))))
Date html generated:
2020_05_20-AM-11_04_31
Last ObjectModification:
2019_12_14-PM-00_54_29
Theory : reals
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