Nuprl Lemma : rational-approx-property2
∀x:ℝ. ∀n:ℕ+. (((x within 1/n) - (r1/r(n))) ≤ x)
Proof
Definitions occuring in Statement :
rational-approx: (x within 1/n),
rdiv: (x/y),
rleq: x ≤ y,
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],
member: t ∈ T,
and: P ∧ Q,
real: ℝ,
top: Top,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
prop: ℙ,
rge: x ≥ y,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
rdiv: (x/y)
Lemmas referenced :
rational-approx-property-ext,
nat_plus_wf,
real_wf,
rminus_wf,
rsub_wf,
rational-approx_wf,
rabs_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_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_formula_prop_wf,
rless_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
uiff_transitivity,
rleq_wf,
radd_wf,
radd-preserves-rleq,
rleq_functionality,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermMinus_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
req_transitivity,
rinv_wf2,
itermMultiply_wf,
real_term_value_mul_lemma,
rmul_wf,
radd_functionality,
req_weakening,
rinv-as-rdiv,
rleq-implies-rleq,
rabs-as-rmax,
rleq-rmax
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
rename,
setElimination,
voidEquality,
voidElimination,
isect_memberEquality,
isectElimination,
lemma_by_obid,
sqequalRule,
because_Cache,
natural_numberEquality,
independent_isectElimination,
inrFormation,
independent_functionElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}x:\mBbbR{}. \mforall{}n:\mBbbN{}\msupplus{}. (((x within 1/n) - (r1/r(n))) \mleq{} x)
Date html generated:
2017_10_03-AM-08_40_52
Last ObjectModification:
2017_07_28-AM-07_31_39
Theory : reals
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