Nuprl Lemma : rroot-exists-part2
∀i:{2...}. ∀x:{x:ℝ| (↑isEven(i)) ⇒ (r0 ≤ x)} . ∀q:{q:ℕ ⟶ ℝ|
(∀n,m:ℕ.
(((r0 ≤ (q n)) ∧ (r0 ≤ (q m))) ∨ (((q n) ≤ r0) ∧ ((q m) ≤ r0))))
∧ ((↑isEven(i)) ⇒ (∀m:ℕ. (r0 ≤ (q m))))} .
(lim n→∞.q n^i = x ⇒ cauchy(n.q n))
Proof
Definitions occuring in Statement :
cauchy: cauchy(n.x[n]),
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],
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],
implies: P ⇒ Q,
converges-to: lim n→∞.x[n] = y,
exists: ∃x:A. B[x],
cauchy: cauchy(n.x[n]),
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
le: A ≤ B,
false: False,
not: ¬A,
so_apply: x[s],
or: P ∨ Q,
int_upper: {i...},
sq_exists: ∃x:{A| B[x]},
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
sq_stable: SqStable(P),
rleq: x ≤ y,
rnonneg: rnonneg(x),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rless: x < y,
real: ℝ
Lemmas referenced :
exp_wf_nat_plus,
mul_nat_plus,
less_than_wf,
converges-to_wf,
rnexp_wf,
int_upper_subtype_nat,
false_wf,
le_wf,
nat_wf,
set_wf,
real_wf,
all_wf,
or_wf,
rleq_wf,
int-to-real_wf,
assert_wf,
isEven_wf,
int_upper_wf,
sq_exists_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
rless-int,
nat_properties,
nat_plus_properties,
int_upper_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
nat_plus_wf,
less_than'_wf,
squash_wf,
sq_stable__all,
sq_stable__rleq,
equal_wf,
exp-fastexp,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
equal-wf-T-base,
exp_wf2,
r-triangle-inequality2,
radd_wf,
exp-positive,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
mul_bounds_1b,
radd_functionality_wrt_rleq,
rleq_functionality,
radd_functionality,
rabs-difference-symmetry,
req_weakening,
rleq-int-fractions,
decidable__le,
intformle_wf,
itermAdd_wf,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
req_transitivity,
radd-rdiv,
rdiv_functionality,
radd-int,
rnexp-convex3,
rnexp-rdiv,
rnexp-positive,
req_functionality,
rnexp-int,
req_wf,
true_wf,
rneq_wf,
exp-one,
iff_weakening_equal,
rleq_transitivity,
rleq_weakening,
rnexp-rleq-iff,
zero-rleq-rabs,
rleq-int-fractions2,
sq_stable__less_than
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
hypothesis,
promote_hyp,
thin,
productElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
sqequalRule,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
setElimination,
rename,
functionEquality,
productEquality,
functionExtensionality,
independent_isectElimination,
inrFormation,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
minusEquality,
independent_pairEquality,
axiomEquality,
imageElimination,
applyLambdaEquality,
dependent_set_memberFormation,
multiplyEquality,
addEquality,
universeEquality
Latex:
\mforall{}i:\{2...\}. \mforall{}x:\{x:\mBbbR{}| (\muparrow{}isEven(i)) {}\mRightarrow{} (r0 \mleq{} x)\} . \mforall{}q:\{q:\mBbbN{} {}\mrightarrow{} \mBbbR{}|
(\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))))\} .
(lim n\mrightarrow{}\minfty{}.q n\^{}i = x {}\mRightarrow{} cauchy(n.q n))
Date html generated:
2017_10_03-AM-10_39_01
Last ObjectModification:
2017_07_28-AM-08_15_31
Theory : reals
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