Nuprl Lemma : near-root-rational-ext
∀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 :
member: t ∈ T,
eq_int: (i =z j),
btrue: tt,
it: ⋅,
bfalse: ff,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
subtract: n - m,
lt_int: i <z j,
near-root: near-root(k;p;q;n),
near-root-rational,
iroot-lemma2,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
uimplies: b supposing a,
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
or: P ∨ Q,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
near-root-rational,
lifting-strict-decide,
istype-void,
has-value_wf_base,
istype-base,
is-exception_wf,
istype-universe,
lifting-strict-int_eq,
strict4-decide,
lifting-strict-callbyvalue,
strict4-spread,
lifting-strict-less,
iroot-lemma2
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination,
independent_pairFormation,
lambdaFormation_alt,
callbyvalueCallbyvalue,
callbyvalueReduce,
universeIsType,
baseApply,
closedConclusion,
hypothesisEquality,
callbyvalueExceptionCases,
inrFormation_alt,
imageMemberEquality,
imageElimination,
exceptionSqequal,
inlFormation_alt,
because_Cache
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:
2019_10_30-AM-07_53_39
Last ObjectModification:
2019_04_02-AM-10_58_22
Theory : reals
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