Nuprl Lemma : approx-root-property
∀k:{2...}. ∀a:{a:ℚ| (0 ≤ a) ∨ (↑isOdd(k))} . ∀n:ℕ+.
((0 ≤ a ⇐⇒ 0 ≤ k-th root(a) within 1/n) ∧ |k-th root(a) within 1/n ↑ k - a| < (1/n))
Proof
Definitions occuring in Statement :
approx-root: k-th root(q) within 1/err,
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],
iff: P ⇐⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
approx-root: k-th root(q) within 1/err,
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
int_upper: {i...},
or: P ∨ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
nat_plus: ℕ+,
so_apply: x[s],
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
guard: {T},
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
sq_exists: ∃x:A [B[x]],
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
qroot-ext,
subtype_rel_self,
int_upper_wf,
all_wf,
rationals_wf,
or_wf,
qle_wf,
assert_wf,
isOdd_wf,
nat_plus_wf,
sq_exists_wf,
iff_wf,
qless_wf,
qabs_wf,
qsub_wf,
qexp_wf,
upper_subtype_nat,
qdiv_wf,
subtype_rel_set,
less_than_wf,
int-subtype-rationals,
int_nzero-rational,
subtype_rel_sets,
nequal_wf,
int_upper_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
equal_wf,
set_wf,
squash_wf,
sq_stable__and,
sq_stable__iff,
sq_stable_from_decidable,
decidable__qle,
decidable__qless,
qless_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
introduction,
sqequalHypSubstitution,
isectElimination,
functionEquality,
natural_numberEquality,
setEquality,
because_Cache,
hypothesisEquality,
setElimination,
rename,
lambdaEquality,
productEquality,
independent_isectElimination,
independent_pairFormation,
intEquality,
productElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
baseClosed,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
imageElimination
Latex:
\mforall{}k:\{2...\}. \mforall{}a:\{a:\mBbbQ{}| (0 \mleq{} a) \mvee{} (\muparrow{}isOdd(k))\} . \mforall{}n:\mBbbN{}\msupplus{}.
((0 \mleq{} a \mLeftarrow{}{}\mRightarrow{} 0 \mleq{} k-th root(a) within 1/n) \mwedge{} |k-th root(a) within 1/n \muparrow{} k - a| < (1/n))
Date html generated:
2018_05_22-AM-00_29_01
Last ObjectModification:
2018_05_19-PM-04_09_13
Theory : rationals
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