Nuprl Lemma : regular-int-seq-iff
∀k:ℕ+. ∀x:ℕ+ ⟶ ℤ.
(k-regular-seq(x)
⇐⇒ ∀n,m:ℕ+.
∃z:ℝ
((((r((x n) - 2 * k)/r((2 * k) * n)) ≤ z) ∧ (z ≤ (r((x n) + (2 * k))/r((2 * k) * n))))
∧ ((r((x m) - 2 * k)/r((2 * k) * m)) ≤ z)
∧ (z ≤ (r((x m) + (2 * k))/r((2 * k) * m)))))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
int-to-real: r(n),
real: ℝ,
regular-int-seq: k-regular-seq(f),
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
so_apply: x[s],
cand: A c∧ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
let: let,
subtype_rel: A ⊆r B,
nat: ℕ,
nequal: a ≠ b ∈ T ,
regular-int-seq: k-regular-seq(f),
uiff: uiff(P;Q),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b
Lemmas referenced :
nat_plus_wf,
regular-int-seq_wf,
all_wf,
exists_wf,
real_wf,
rleq_wf,
rdiv_wf,
int-to-real_wf,
subtract_wf,
rless-int,
multiply_nat_plus,
less_than_wf,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermMultiply_wf,
intformeq_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_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
equal_wf,
rless_wf,
regular-iff-all-regular-upto,
imax_wf,
imax_nat_plus,
regular-upto-iff,
imax_ub,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
le_wf,
itermAdd_wf,
int_term_value_add_lemma,
lelt_wf,
seq-min-upper_wf,
nat_plus_subtype_nat,
rneq-int,
int_entire_a,
equal-wf-base,
int_subtype_base,
equal-wf-T-base,
rleq_transitivity,
rleq-int-fractions,
mul_nat_plus,
absval_unfold,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
mul_cancel_in_le,
multiply-is-int-iff,
add-is-int-iff,
subtract-is-int-iff,
itermSubtract_wf,
int_term_value_subtract_lemma,
false_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
itermMinus_wf,
int_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
functionExtensionality,
applyEquality,
sqequalRule,
lambdaEquality,
because_Cache,
productEquality,
multiplyEquality,
natural_numberEquality,
independent_isectElimination,
inrFormation,
dependent_functionElimination,
productElimination,
independent_functionElimination,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
unionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
addEquality,
functionEquality,
inlFormation,
baseApply,
closedConclusion,
promote_hyp,
minusEquality,
equalityElimination,
lessCases,
isect_memberFormation,
sqequalAxiom,
imageElimination,
pointwiseFunctionality,
instantiate
Latex:
\mforall{}k:\mBbbN{}\msupplus{}. \mforall{}x:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}.
(k-regular-seq(x)
\mLeftarrow{}{}\mRightarrow{} \mforall{}n,m:\mBbbN{}\msupplus{}.
\mexists{}z:\mBbbR{}
((((r((x n) - 2 * k)/r((2 * k) * n)) \mleq{} z) \mwedge{} (z \mleq{} (r((x n) + (2 * k))/r((2 * k) * n))))
\mwedge{} ((r((x m) - 2 * k)/r((2 * k) * m)) \mleq{} z)
\mwedge{} (z \mleq{} (r((x m) + (2 * k))/r((2 * k) * m)))))
Date html generated:
2017_10_03-AM-08_44_57
Last ObjectModification:
2017_09_11-PM-01_33_13
Theory : reals
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