Nuprl Lemma : approx-root_wf
∀k:{2...}. ∀a:{a:ℚ| (0 ≤ a) ∨ (↑isOdd(k))} . ∀n:ℕ+. (k-th root(a) within 1/n ∈ ℚ)
Proof
Definitions occuring in Statement :
approx-root: k-th root(q) within 1/err,
qle: r ≤ s,
rationals: ℚ,
isOdd: isOdd(n),
int_upper: {i...},
nat_plus: ℕ+,
assert: ↑b,
all: ∀x:A. B[x],
or: P ∨ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
approx-root: k-th root(q) within 1/err,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
nat_plus: ℕ+,
so_apply: x[s],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
int_nzero: ℤ-o,
implies: P ⇒ Q,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
guard: {T},
int_upper: {i...},
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
sq_exists: ∃x:A [B[x]],
or: P ∨ Q
Lemmas referenced :
qroot-ext,
subtype_rel_self,
all_wf,
sq_exists_wf,
qless_wf,
qdiv_wf,
subtype_rel_set,
rationals_wf,
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,
or_wf,
qle_wf,
assert_wf,
isOdd_wf,
nat_plus_wf,
set_wf,
int_upper_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
setElimination,
thin,
rename,
sqequalRule,
applyEquality,
instantiate,
extract_by_obid,
hypothesis,
introduction,
sqequalHypSubstitution,
isectElimination,
functionEquality,
because_Cache,
lambdaEquality,
productEquality,
hypothesisEquality,
intEquality,
natural_numberEquality,
independent_isectElimination,
productElimination,
setEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
baseClosed,
dependent_set_memberEquality
Latex:
\mforall{}k:\{2...\}. \mforall{}a:\{a:\mBbbQ{}| (0 \mleq{} a) \mvee{} (\muparrow{}isOdd(k))\} . \mforall{}n:\mBbbN{}\msupplus{}. (k-th root(a) within 1/n \mmember{} \mBbbQ{})
Date html generated:
2018_05_22-AM-00_28_55
Last ObjectModification:
2018_05_19-PM-04_08_52
Theory : rationals
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