Nuprl Lemma : implies-regular
∀[k:ℕ+]. ∀[x:ℕ+ ⟶ ℤ].
k-regular-seq(x) supposing ∀n,m:ℕ+. (|(x within 1/n) - (x within 1/m)| ≤ ((r(k)/r(n)) + (r(k)/r(m))))
Proof
Definitions occuring in Statement :
rational-approx: (x within 1/n),
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
radd: a + b,
int-to-real: r(n),
regular-int-seq: k-regular-seq(f),
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
rational-approx: (x within 1/n),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
regular-int-seq: k-regular-seq(f),
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
so_lambda: λ2x.t[x],
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
so_apply: x[s],
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
uiff: uiff(P;Q),
real: ℝ,
sq_stable: SqStable(P),
rev_uimplies: rev_uimplies(P;Q),
sq_type: SQType(T),
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
rleq: x ≤ y,
rnonneg: rnonneg(x)
Lemmas referenced :
less_than'_wf,
absval_wf,
subtract_wf,
nat_plus_wf,
nat_wf,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
int-rdiv_wf,
nat_plus_properties,
satisfiable-full-omega-tt,
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,
int-to-real_wf,
radd_wf,
rdiv_wf,
rless-int,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
rless_wf,
mul_bounds_1b,
less_than_wf,
mul_nat_plus,
equal-wf-T-base,
req_wf,
squash_wf,
true_wf,
real_wf,
rabs-int,
iff_weakening_equal,
req-int,
equal_wf,
absval_pos,
mul-non-neg1,
false_wf,
sq_stable__less_than,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
le_wf,
rleq_functionality,
req_weakening,
radd-int-fractions,
rmul_preserves_req,
rmul_wf,
rinv_wf2,
rneq_functionality,
rmul-int,
rneq-int,
rminus_wf,
minus-one-mul,
subtype_base_sq,
itermAdd_wf,
int_term_value_add_lemma,
req_functionality,
rsub_functionality,
int-rdiv-req,
req_transitivity,
rmul_functionality,
rinv_functionality2,
req_inversion,
rinv-of-rmul,
real_term_polynomial,
itermSubtract_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
req-iff-rsub-is-0,
radd_functionality,
rmul-rinv,
rmul-rinv3,
int-rinv-cancel,
itermMinus_wf,
real_term_value_minus_lemma,
rminus_functionality,
rminus-int,
radd-int,
rsub-int,
rabs_functionality,
rless_transitivity1,
rleq_weakening,
rmul_preserves_rleq2,
rleq_weakening_rless,
rabs-rdiv,
rdiv_functionality,
rleq-int
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
lambdaEquality,
productElimination,
independent_pairEquality,
extract_by_obid,
isectElimination,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
addEquality,
applyEquality,
functionExtensionality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
baseApply,
closedConclusion,
baseClosed,
inrFormation,
independent_functionElimination,
unionElimination,
functionEquality,
imageMemberEquality,
imageElimination,
universeEquality,
minusEquality,
addLevel,
instantiate,
cumulativity,
promote_hyp
Latex:
\mforall{}[k:\mBbbN{}\msupplus{}]. \mforall{}[x:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}].
k-regular-seq(x)
supposing \mforall{}n,m:\mBbbN{}\msupplus{}. (|(x within 1/n) - (x within 1/m)| \mleq{} ((r(k)/r(n)) + (r(k)/r(m))))
Date html generated:
2017_10_03-AM-08_51_24
Last ObjectModification:
2017_07_28-AM-07_34_41
Theory : reals
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