Nuprl Lemma : rational-approx-property1
∀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,
radd: a + b,
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,
uall: ∀[x:A]. B[x],
real: ℝ,
and: P ∧ Q,
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,
top: Top,
prop: ℙ,
rge: x ≥ y,
uiff: uiff(P;Q),
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
rsub: x - y
Lemmas referenced :
radd-rminus-assoc,
req_weakening,
radd_comm,
radd_functionality,
rleq_functionality,
uiff_transitivity,
iff_weakening_equal,
radd_comm_eq,
true_wf,
squash_wf,
rminus_wf,
radd_wf,
rleq_wf,
radd-preserves-rleq,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rless_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
rless-int,
int-to-real_wf,
rdiv_wf,
rabs_wf,
rational-approx_wf,
rsub_wf,
rabs-bounds,
real_wf,
nat_plus_wf,
rational-approx-property-ext
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
setElimination,
rename,
productElimination,
natural_numberEquality,
independent_isectElimination,
sqequalRule,
inrFormation,
because_Cache,
independent_functionElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}x:\mBbbR{}. \mforall{}n:\mBbbN{}\msupplus{}. (x \mleq{} ((x within 1/n) + (r1/r(n))))
Date html generated:
2016_05_18-AM-07_30_17
Last ObjectModification:
2016_01_17-AM-02_00_07
Theory : reals
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