Nuprl Lemma : approximate-qsqrt-ext
∀a:{a:ℚ| 0 ≤ a} . ∀n:ℕ+. (∃q:ℚ [((0 ≤ q) ∧ |(q * q) - a| < (1/n))])
Proof
Definitions occuring in Statement :
qabs: |r|,
qle: r ≤ s,
qless: r < s,
qsub: r - s,
qdiv: (r/s),
qmul: r * s,
rationals: ℚ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
experimental: experimental{impliesFunctionality}(possibleextract),
subtract: n - m,
qsub: r - s,
qadd: r + s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
outl: outl(x),
bottom: ⊥,
outr: outr(x),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
bfalse: ff,
qpositive: qpositive(r),
lt_int: i <z j,
bor: p ∨bq,
band: p ∧b q,
qeq: qeq(r;s),
eq_int: (i =z j),
approximate-qsqrt,
better-q-elim,
square-between-lemma3,
sq_stable_from_decidable,
decidable__qle,
q-elim,
square-between-lemma2,
sq_stable__from_stable,
stable__from_decidable,
any: any x,
decidable__lt,
square-between-lemma1,
decidable__squash,
decidable__and,
decidable__less_than',
decidable_functionality,
squash_elim,
iff_preserves_decidability,
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]
Lemmas referenced :
approximate-qsqrt,
lifting-strict-spread,
istype-void,
strict4-apply,
strict4-spread,
lifting-strict-decide,
strict4-decide,
lifting-strict-less,
better-q-elim,
square-between-lemma3,
sq_stable_from_decidable,
decidable__qle,
q-elim,
square-between-lemma2,
sq_stable__from_stable,
stable__from_decidable,
decidable__lt,
square-between-lemma1,
decidable__squash,
decidable__and,
decidable__less_than',
decidable_functionality,
squash_elim,
iff_preserves_decidability
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
Latex:
\mforall{}a:\{a:\mBbbQ{}| 0 \mleq{} a\} . \mforall{}n:\mBbbN{}\msupplus{}. (\mexists{}q:\mBbbQ{} [((0 \mleq{} q) \mwedge{} |(q * q) - a| < (1/n))])
Date html generated:
2019_10_16-PM-00_38_12
Last ObjectModification:
2019_06_26-PM-04_16_38
Theory : rationals
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