Nuprl Lemma : near-root-rational
∀k:{2...}. ∀p:{p:ℤ| (0 ≤ p) ∨ (↑isOdd(k))} . ∀q,n:ℕ+.
(∃r:ℤ × ℕ+ [let a,b = r
in (0 ≤ p ⇐⇒ 0 ≤ a) ∧ (|(r(a))/b^k - (r(p)/r(q))| < (r1/r(n)))])
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rless: x < y,
rabs: |x|,
rnexp: x^k1,
int-rdiv: (a)/k1,
rsub: x - y,
int-to-real: r(n),
isOdd: isOdd(n),
int_upper: {i...},
nat_plus: ℕ+,
assert: ↑b,
le: A ≤ B,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
iff: P ⇐⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
spread: spread def,
product: x:A × B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
or: P ∨ Q,
int_upper: {i...},
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
nat_plus: ℕ+,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
bfalse: ff,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
so_lambda: λ2x.t[x],
so_apply: x[s],
bool: 𝔹,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
nat: ℕ,
unit: Unit,
it: ⋅,
btrue: tt,
bnot: ¬bb,
assert: ↑b,
eq_int: (i =z j),
nequal: a ≠ b ∈ T ,
top: Top,
subtype_rel: A ⊆r B,
subtract: n - m,
ge: i ≥ j ,
le: A ≤ B,
less_than': less_than'(a;b),
exp: i^n,
primrec: primrec(n;b;c),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
sq_exists: ∃x:A [B[x]],
rneq: x ≠ y,
cand: A c∧ B,
less_than: a < b,
rless: x < y,
isOdd: isOdd(n),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
req_int_terms: t1 ≡ t2,
rdiv: (x/y)
Latex:
\mforall{}k:\{2...\}. \mforall{}p:\{p:\mBbbZ{}| (0 \mleq{} p) \mvee{} (\muparrow{}isOdd(k))\} . \mforall{}q,n:\mBbbN{}\msupplus{}.
(\mexists{}r:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{} [let a,b = r
in (0 \mleq{} p \mLeftarrow{}{}\mRightarrow{} 0 \mleq{} a) \mwedge{} (|(r(a))/b\^{}k - (r(p)/r(q))| < (r1/r(n)))])
Date html generated:
2020_05_20-PM-00_29_17
Last ObjectModification:
2020_01_06-PM-00_55_43
Theory : reals
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