Nuprl Lemma : rational-approx-property
∀x:ℝ. ∀n:ℕ+. (|x - (x within 1/n)| ≤ (r1/r(n)))
Proof
Definitions occuring in Statement :
rational-approx: (x within 1/n),
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
rational-approx: (x within 1/n),
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_nzero: ℤ-o,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
real: ℝ,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
true: True,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
nat: ℕ
Lemmas referenced :
nat_plus_wf,
real_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,
rdiv_wf,
rless-int,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
rless_wf,
rmul_preserves_rleq,
rmul_wf,
radd_wf,
rinv_wf2,
rminus_wf,
rleq_functionality,
rabs_functionality,
rsub_functionality,
req_weakening,
int-rdiv-req,
req_transitivity,
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,
req-iff-rsub-is-0,
rmul_functionality,
itermAdd_wf,
itermMinus_wf,
real_term_value_add_lemma,
real_term_value_minus_lemma,
radd_functionality,
rminus-rdiv,
squash_wf,
true_wf,
rneq_wf,
rminus-int,
rmul-rinv,
rmul-int,
rmul-assoc,
req_wf,
rabs-int,
iff_weakening_equal,
req-int,
absval_wf,
nat_wf,
equal_wf,
absval_pos,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
le_wf,
req_inversion,
rabs-rmul,
real-approx
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
because_Cache,
inrFormation,
productElimination,
independent_functionElimination,
unionElimination,
minusEquality,
imageMemberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality
Latex:
\mforall{}x:\mBbbR{}. \mforall{}n:\mBbbN{}\msupplus{}. (|x - (x within 1/n)| \mleq{} (r1/r(n)))
Date html generated:
2017_10_03-AM-08_40_44
Last ObjectModification:
2017_07_28-AM-07_31_25
Theory : reals
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