Nuprl Lemma : rational-inner-approx-property
∀x:ℝ. ∀n:ℕ+. ((|rational-inner-approx(x;n)| ≤ |x|) ∧ (|x - rational-inner-approx(x;n)| ≤ (r(2)/r(n))))
Proof
Definitions occuring in Statement :
rational-inner-approx: rational-inner-approx(x;n),
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],
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
rational-approx: (x within 1/n),
rational-inner-approx: rational-inner-approx(x;n),
real: ℝ,
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),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
has-value: (a)↓,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
rneq: x ≠ y,
top: Top,
rev_uimplies: rev_uimplies(P;Q),
rat_term_to_real: rat_term_to_real(f;t),
rtermSubtract: left "-" right,
rat_term_ind: rat_term_ind,
rtermDivide: num "/" denom,
rtermConstant: "const",
rtermVar: rtermVar(var),
pi1: fst(t),
true: True,
pi2: snd(t),
rless: x < y,
sq_exists: ∃x:A [B[x]],
int-to-real: r(n),
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
req_int_terms: t1 ≡ t2,
sq_stable: SqStable(P),
squash: ↓T,
nat: ℕ,
rge: x ≥ y,
absval: |i|,
rtermAdd: left "+" right,
less_than: a < b,
less_than': less_than'(a;b),
rdiv: (x/y),
ge: i ≥ j
Latex:
\mforall{}x:\mBbbR{}. \mforall{}n:\mBbbN{}\msupplus{}.
((|rational-inner-approx(x;n)| \mleq{} |x|) \mwedge{} (|x - rational-inner-approx(x;n)| \mleq{} (r(2)/r(n))))
Date html generated:
2020_05_20-AM-11_04_16
Last ObjectModification:
2020_01_03-PM-08_04_47
Theory : reals
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