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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