Nuprl Lemma : rroot-exists-part1
∀i:{2...}. ∀x:{x:ℝ| (↑isEven(i)) ⇒ (r0 ≤ x)} .
∃q:ℕ ⟶ ℝ
(lim n→∞.q n^i = x
∧ (∀n,m:ℕ. (((r0 ≤ (q n)) ∧ (r0 ≤ (q m))) ∨ (((q n) ≤ r0) ∧ ((q m) ≤ r0))))
∧ ((↑isEven(i)) ⇒ (∀m:ℕ. (r0 ≤ (q m)))))
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
rleq: x ≤ y,
rnexp: x^k1,
int-to-real: r(n),
real: ℝ,
isEven: isEven(n),
int_upper: {i...},
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
int_upper: {i...},
real: ℝ,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
false: False,
not: ¬A,
or: P ∨ Q,
rational-approx: (x within 1/n),
top: Top,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
guard: {T},
satisfiable_int_formula: satisfiable_int_formula(fmla),
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
uiff: uiff(P;Q),
nat: ℕ,
rless: x < y,
sq_exists: ∃x:A [B[x]],
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
sq_stable: SqStable(P),
cand: A c∧ B,
subtract: n - m,
pi1: fst(t),
converges-to: lim n→∞.x[n] = y,
ge: i ≥ j
Lemmas referenced :
equal_wf,
set-value-type,
real_wf,
assert_wf,
isEven_wf,
rleq_wf,
int-to-real_wf,
nat_plus_wf,
regular-int-seq_wf,
function-value-type,
less_than_wf,
int-value-type,
value-type_wf,
exists_wf,
nat_wf,
converges-to_wf,
rnexp_wf,
upper_subtype_nat,
false_wf,
all_wf,
or_wf,
set_wf,
int_upper_wf,
rational-approx-property,
mul_nat_plus,
rabs_wf,
rsub_wf,
int-rdiv_wf,
nat_plus_properties,
int_upper_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
nequal_wf,
rdiv_wf,
rless-int,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
rless_wf,
mul-associates,
rleq_functionality,
rabs_functionality,
rsub_functionality,
req_weakening,
int-rdiv-req,
squash_wf,
true_wf,
rneq_wf,
subtype_rel_self,
iff_weakening_equal,
decidable__or,
decidable__rless-int-fractions,
decidable__le,
rleq-int-fractions2,
intformle_wf,
int_formula_prop_le_lemma,
isOdd_wf,
odd-or-even,
assert_of_bor,
rabs-difference-bound-rleq,
mul_preserves_le,
le_wf,
radd-preserves-rless,
rless-int-fractions,
radd_wf,
rmul_wf,
rinv_wf2,
rneq_functionality,
rmul-int,
rneq-int,
equal-wf-T-base,
itermSubtract_wf,
itermAdd_wf,
rless_transitivity1,
rless_irreflexivity,
rless_functionality,
req_transitivity,
radd_functionality,
rmul_functionality,
rinv_functionality2,
req_inversion,
rinv-of-rmul,
rinv-mul-as-rdiv,
rdiv_functionality,
rinv-as-rdiv,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
iff_wf,
subtype_rel_sets,
near-root_wf,
zero-mul,
product-value-type,
req_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
r-triangle-inequality2,
rabs-difference-symmetry,
radd_functionality_wrt_rleq,
rleq_weakening_rless,
radd_functionality_wrt_rless2,
rleq-int-fractions,
sq_stable__less_than,
int_term_value_add_lemma,
radd-rdiv,
radd-int,
sq_stable__le,
intformimplies_wf,
int_formual_prop_imp_lemma,
rmul_preserves_rless,
rmul-rinv,
rless-implies-rless,
mul_preserves_lt,
rless_transitivity2,
req_functionality,
decidable__equal_int,
intformor_wf,
int_formula_prop_or_lemma,
rmul_preserves_req,
req-int,
exp_wf2,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
rminus_wf,
itermMinus_wf,
rabs-rminus,
rnexp-int,
exp-zero,
real_term_value_minus_lemma,
subtract_wf,
rsub_functionality_wrt_rleq,
rsub-rdiv,
rsub-int,
rabs-as-rmax,
rmax_lb,
rminus-as-rmul,
rmul-assoc,
rminus-reverses-rleq,
square-nonzero,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-associates,
add-zero,
int_term_value_subtract_lemma,
less-iff-le,
minus-minus,
add-swap,
nat_properties,
rleq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
because_Cache,
hypothesisEquality,
cutEval,
introduction,
dependent_set_memberEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
hypothesis,
lambdaEquality,
independent_isectElimination,
functionEquality,
setElimination,
rename,
natural_numberEquality,
intEquality,
dependent_pairFormation,
independent_pairFormation,
imageMemberEquality,
baseClosed,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
productEquality,
applyEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
multiplyEquality,
approximateComputation,
independent_functionElimination,
int_eqEquality,
baseApply,
closedConclusion,
inrFormation,
productElimination,
unionElimination,
equalityTransitivity,
minusEquality,
imageElimination,
instantiate,
universeEquality,
inlFormation,
setEquality,
promote_hyp,
addEquality,
addLevel,
functionExtensionality,
dependent_set_memberFormation
Latex:
\mforall{}i:\{2...\}. \mforall{}x:\{x:\mBbbR{}| (\muparrow{}isEven(i)) {}\mRightarrow{} (r0 \mleq{} x)\} .
\mexists{}q:\mBbbN{} {}\mrightarrow{} \mBbbR{}
(lim n\mrightarrow{}\minfty{}.q n\^{}i = x
\mwedge{} (\mforall{}n,m:\mBbbN{}. (((r0 \mleq{} (q n)) \mwedge{} (r0 \mleq{} (q m))) \mvee{} (((q n) \mleq{} r0) \mwedge{} ((q m) \mleq{} r0))))
\mwedge{} ((\muparrow{}isEven(i)) {}\mRightarrow{} (\mforall{}m:\mBbbN{}. (r0 \mleq{} (q m)))))
Date html generated:
2019_10_30-AM-07_54_15
Last ObjectModification:
2018_08_27-PM-11_52_17
Theory : reals
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