Nuprl Lemma : qlog-lemma-ext
∀e:{e:ℚ| 0 < e} . ∀q:{q:ℚ| (e ≤ q) ∧ q < 1} . {nr:ℕ × ℚ| let n,r = nr in (r = q ↑ n ∈ ℚ) ∧ (e ≤ r) ∧ r * r < e}
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qle: r ≤ s,
qless: r < s,
qmul: r * s,
rationals: ℚ,
nat: ℕ,
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
spread: spread def,
product: x:A × B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
qlog-lemma,
uniform-comp-nat-induction,
decidable__qless,
ifthenelse: if b then t else f fi ,
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,
guard: {T},
or: P ∨ Q,
squash: ↓T,
btrue: tt,
genrec-ap: genrec-ap
Lemmas referenced :
qlog-lemma,
lifting-strict-decide,
top_wf,
equal_wf,
has-value_wf_base,
base_wf,
is-exception_wf,
uniform-comp-nat-induction,
decidable__qless
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
isectElimination,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
callbyvalueDecide,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
unionEquality,
unionElimination,
sqleReflexivity,
dependent_functionElimination,
independent_functionElimination,
baseApply,
closedConclusion,
decideExceptionCases,
inrFormation,
because_Cache,
imageMemberEquality,
imageElimination,
exceptionSqequal,
inlFormation
Latex:
\mforall{}e:\{e:\mBbbQ{}| 0 < e\} . \mforall{}q:\{q:\mBbbQ{}| (e \mleq{} q) \mwedge{} q < 1\} .
\{nr:\mBbbN{} \mtimes{} \mBbbQ{}| let n,r = nr in (r = q \muparrow{} n) \mwedge{} (e \mleq{} r) \mwedge{} r * r < e\}
Date html generated:
2018_05_22-AM-00_11_07
Last ObjectModification:
2017_07_26-PM-06_52_29
Theory : rationals
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