Nuprl Lemma : qroot-ext
∀k:{2...}. ∀a:{a:ℚ| (0 ≤ a) ∨ (↑isOdd(k))} . ∀n:ℕ+. (∃q:ℚ [((0 ≤ a ⇐⇒ 0 ≤ q) ∧ |q ↑ k - a| < (1/n))])
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qabs: |r|,
qle: r ≤ s,
qless: r < s,
qsub: r - s,
qdiv: (r/s),
rationals: ℚ,
isOdd: isOdd(n),
int_upper: {i...},
nat_plus: ℕ+,
assert: ↑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]} ,
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
experimental: experimental{impliesFunctionality}(possibleextract),
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,
absval: |i|,
divide: n ÷ m,
lt_int: i <z j,
isOdd: isOdd(n),
modulus: a mod n,
remainder: n rem m,
qsub: r - s,
qadd: r + s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
outl: outl(x),
bottom: ⊥,
outr: outr(x),
qpositive: qpositive(r),
bor: p ∨bq,
qeq: qeq(r;s),
qroot,
better-q-elim,
iroot-lemma2,
sq_stable_from_decidable,
decidable__or,
decidable__qle,
decidable__assert,
q-elim,
sq_stable__from_stable,
stable__from_decidable,
any: any x,
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],
top: Top,
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
or: P ∨ Q,
squash: ↓T
Lemmas referenced :
qroot,
lifting-strict-spread,
istype-void,
strict4-apply,
strict4-spread,
lifting-strict-decide,
has-value_wf_base,
istype-base,
is-exception_wf,
istype-universe,
lifting-strict-int_eq,
strict4-decide,
lifting-strict-callbyvalue,
value-type-has-value,
int-value-type,
lifting-strict-less,
better-q-elim,
iroot-lemma2,
sq_stable_from_decidable,
decidable__or,
decidable__qle,
decidable__assert,
q-elim,
sq_stable__from_stable,
stable__from_decidable
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,
callbyvalueAdd,
productElimination,
intEquality,
addExceptionCases,
callbyvalueMultiply,
multiplyExceptionCases
Latex:
\mforall{}k:\{2...\}. \mforall{}a:\{a:\mBbbQ{}| (0 \mleq{} a) \mvee{} (\muparrow{}isOdd(k))\} . \mforall{}n:\mBbbN{}\msupplus{}.
(\mexists{}q:\mBbbQ{} [((0 \mleq{} a \mLeftarrow{}{}\mRightarrow{} 0 \mleq{} q) \mwedge{} |q \muparrow{} k - a| < (1/n))])
Date html generated:
2019_10_16-PM-00_37_45
Last ObjectModification:
2019_06_26-PM-04_15_04
Theory : rationals
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