Nuprl Lemma : rational-approx-property-alt
∀x:ℝ. ∀n:ℕ+. (|(r(2 * n) * x) - r(x n)| ≤ r(2))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
apply: f a,
multiply: n * m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
real: ℝ,
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
le: A ≤ B,
less_than': less_than'(a;b),
rdiv: (x/y),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rational-approx: (x within 1/n),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
true: True,
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T)
Lemmas referenced :
rational-approx-property,
rmul_preserves_rleq2,
rabs_wf,
rsub_wf,
rational-approx_wf,
int-to-real_wf,
rdiv_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
rleq-int,
decidable__le,
intformle_wf,
itermMultiply_wf,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
nat_plus_wf,
real_wf,
rmul_wf,
radd_wf,
rminus_wf,
itermSubtract_wf,
itermAdd_wf,
itermMinus_wf,
rinv_wf2,
rleq_wf,
istype-false,
rleq_functionality,
req_transitivity,
rmul_functionality,
req_weakening,
req_inversion,
rmul-int,
rabs_functionality,
rmul-rinv,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
uiff_transitivity,
rmul-nonneg-rabs,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening,
int-rdiv_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
int_subtype_base,
nequal_wf,
rneq_functionality,
rneq-int,
set_subtype_base,
less_than_wf,
minus-one-mul-top,
subtype_base_sq,
req_functionality,
radd_functionality,
rminus_functionality,
int-rdiv-req,
rsub_functionality,
squash_wf,
true_wf,
rminus-int,
rinv_functionality2,
rinv-of-rmul,
int-rinv-cancel
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
setElimination,
rename,
multiplyEquality,
natural_numberEquality,
closedConclusion,
because_Cache,
independent_isectElimination,
sqequalRule,
inrFormation_alt,
productElimination,
independent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
applyEquality,
dependent_set_memberEquality_alt,
equalityIstype,
inhabitedIsType,
baseApply,
baseClosed,
sqequalBase,
equalitySymmetry,
intEquality,
minusEquality,
imageMemberEquality,
instantiate,
cumulativity,
equalityTransitivity,
imageElimination
Latex:
\mforall{}x:\mBbbR{}. \mforall{}n:\mBbbN{}\msupplus{}. (|(r(2 * n) * x) - r(x n)| \mleq{} r(2))
Date html generated:
2019_10_29-AM-10_00_45
Last ObjectModification:
2019_02_13-PM-02_44_43
Theory : reals
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