Nuprl Lemma : rational-approx-property-alt2
∀x:ℝ. ∀n:ℕ+. (|r(x n)| ≤ ((r(2 * n) * |x|) + r(2)))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rabs: |x|,
rmul: a * b,
radd: 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],
nat_plus: ℕ+,
real: ℝ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
req_int_terms: t1 ≡ t2,
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
rge: x ≥ y,
guard: {T}
Lemmas referenced :
rational-approx-property-alt,
nat_plus_wf,
real_wf,
rabs_wf,
rsub_wf,
rmul_wf,
int-to-real_wf,
radd_wf,
itermSubtract_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
req-iff-rsub-is-0,
rleq_functionality,
rabs-difference-symmetry,
req_weakening,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
rabs_functionality,
rleq-int,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
rleq_weakening,
rleq_functionality_wrt_implies,
r-triangle-inequality,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
radd_functionality,
rabs-rmul,
req_functionality,
rabs-of-nonneg,
rmul_functionality
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
universeIsType,
isectElimination,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
applyEquality,
because_Cache,
productElimination,
independent_isectElimination,
sqequalRule,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
voidElimination,
independent_functionElimination,
unionElimination,
dependent_pairFormation_alt,
independent_pairFormation
Latex:
\mforall{}x:\mBbbR{}. \mforall{}n:\mBbbN{}\msupplus{}. (|r(x n)| \mleq{} ((r(2 * n) * |x|) + r(2)))
Date html generated:
2019_10_29-AM-10_00_57
Last ObjectModification:
2019_02_14-PM-07_13_43
Theory : reals
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